^ 


liversity  of  Califorr. 

FROM    THK    I.HiKAkV    n}- 

DR.    FRANCIS     LIEBER, 
pv.,,f.,c;,.,  of  irit^trry  and  Law  in  Columbia  College.  New 


THE  GU'T  01' 

MICHAEL    REESE 

Of  San  Francisci 
1873. 


i 


Digitized  by  tlie  Internet  Arcliive 

in  2008  witli  funding  from 

IVIicrosoft  Corporation 


littp://www.archive.org/details/elementsofalgebrOOdaviricli 


77 


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ELEMENTS 


ALGEBRA 


TRANSLATED  FROM  THE  FRENCH  OF 


M.    BOURDON. 


REVISED  AND  ADATTED  TO  THE  COURSES  OP  MATHEMATICAL  INSTRUCTION 
IN  THE  UNITED  STATES ; 

BY    CHARI^ES    DAYIES, 

PROFESSOR   OF    MATHEMATICS    IN    THE    MILITARY    ACADEMY 

AND 

AUTHOR    OF    THE    COMMON    SCHOOL    ARITHMETIC,    ELEMENTS    OF 

DESCRIPTIVE  GEOMETRY,  8VRVEYING,  AND  A  TREATISE 

ON  SHADOWS  AND  PERSPECTIVE. 


NEW.  YORK  : 

PUBLISHED   BY   WILEY   &   LONG, 
No.  161  Broadway. 

•ITSr.EOTVPCD  BY  A.  CIIA*<DLCR. 

1835. 


Entered  according  to  the  Act  of  Congress,  in  the  year  one  thousand  eight 
hundred  and  thirty-five,  by  Charles  Davies,  in  the  Clerk's  Office  of  the  Dis- 
trict Court  of  the  United  States,  for  the  Southern  District  of  New- York. 


Cy^^M"^"^ 


^^ 


PREFACE. 


TiiK  Treatise  on  Algebra,  by  Bourdon,  is  a  work  of  sin- 
gular excellence  and  merit.  In  France,  it  is  one  of  the 
leading  text  books,  and  shortly  after  its  publication,  had 
passed  through  several  editions.  It  has  been  translated,  in 
part,  by  Professor  De  Morgan,  of  the  London  University, 
and  it  is  now^  used  in  the  University  of  Cambridge. 

A  translation  v^^as  made  by  Lt.  Ross,  and  published  in 
1831,  since  which  time  it  has  been  adopted  as  a  text  book 
in  the  Military  Academy,  the  University  of  the  City  of  New- 
York,  Union  College,  Princeton  College,  Geneva  College, 
and  in  Kenyon  College,  in  Ohio. 

The  original  work  is  a  full  and  complete  treatise  on  the 
subject  of  Algebra,  and  contains  six  hundred  and  seventy 
pages  octavo.  The  time  which  is  given  to  the  study  of  Al- 
gebra, even  in  those  seminaries  where  the  course  of  mathe- 
matics is  the  fullest,  is  too  short  to  accomplish  so  voluminous 
a  work,  and  hence  it  has  been  found  necessary  either  to 
modify  it,  or  abandon  it  altogether. 


6^0  a 


PREFACE. 


The  work  which  is  here  presented  to  the  public,  is  au 
abridgment  of  Bourdon ;  with  such  modifications,  as  expe- 
rience in  teaching  it,  and  a  very  careful  comparison  with 
other  standard  works,  have  suggested. 

It  has  been  the  intention  to  unite  in  this  work,  the  scien- 
tific discussions  of  the  French,  with  the  practical  methods  of 
the  English  school ;  that  theory  and  practice,  science  and 
art,  may  mutually  aid  and  illusti-ate  each  other. 

Military  Academy,  March,  1835. 


CONTENTS. 


CHAPTER  I. 

Preliminary  Definition  and  Remarks. 

Algebra — Definitions — Explanation  of  the  Algebraic  Signs, 

Similar  Terms — Reduction  of  Similar  Terms, 

Problems — Theorems — Definition  of— Question, 

Addition — Rule, 

Subtraction — Rule — Remark, 

Multiplication — Rule  for  Monomials, 

Rule  for  Polynomials  and  Signs, 

Remarks — Theorems  Proved, 

Division  of  Monomials — Rule, 

Signification  of  the  Symbol  a*, 

Division  of  Polynomials — Rule, 

When  771  is  entire,  a'^-b'^  is  divisible  by  a-b, 

Remarks,        ..... 


ARTICLES. 

1—28 
28—30 
31—33 
33—36 
36—40 
40—42 
42—45 
45—49 
49—52 
52—54 
54—59 
59—60 
60—62 


ALGEBRAIC    FRACTIONS. 

Definition — Entire  Quantity — Mixed  Quantity, 

Reduction  of  Fractions, 

Greatest  Common  Divisor — Theory  of. 

To  Reduce  a  Mixed  Quantity  to  a  Fraction, 

To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity, 

To  Reduce  Fractions  to  a  Common  Denominator, 

To  Add  Fractions,       .  •  .  . 

To  Subtract  Fractions, 

To  Multiply  Fractions, 

To  Divide  Fractions,  .... 


65—66 

66—71 

71 

72 

73 

•      ^^ 
75 

76 

77 


CHAPTER  II. 
Equations  of  the  First  Degree. 

Definition  of  an  Equation — Different  Kinds — Properties  of 

Equations,  .... 

Equations  involving  but  One  Unknown  Quantity, 
Transformation  of  Equations — First  and  Second, 
Resolution  of  Equations  of  the  First  Degree — Rule, 
Questions  involving  Equations  of  the  First  Degree, 
Equations  of  the  First  Degree  involving  Two  Unknown 

Quantities,  ..... 

Elimination — By  Addition — By  Subtraction — By  Comparison. 
Resolution  of  Questions  involving  Two  or  more  Unknown 

Quantities,  ..... 

Theory  of  Negative  Quantities — Explanation  of  the  Terms 

Nothing  and  Infinity,       .... 
Inequalities,     ...... 


79-86 
86—87 
87—92 
92—94 
94—95 

95—96 
96—103 

103—104 

104—114 
114—116 


CHAPTER  III. 


Extraction  of  the  Square  Root  of  Numbers,  .            .            .  116 — 119 

Extraction  of  the  Square  Root  of  Fractions,  .            .             .  119 — 124 
Extraction  of  the  Square  Root  of  Algebraic  Quantities — 

Of  Monomials,     ......  124—127 

Of  Polynomifils,           ......  127—130 

Calculus  of  Radicals  of  the  Second  Degree,  .             .  130 — 132 

Addition  and  Subtraction— Of  Radicals,          .             .             .  132 — 133 

Multiplication,  Division  and  Transformation,  .             .             .  133 — 137 

Equations  of  the  Second  Degree,         ....  137 — 139 

Involving  Two  Tenns,            .....  139 — 140 

Complete  Equations  of  the  Second  Degree,     .             .             .  140 — 141 

Discus^on  of  Equations  of  the  Second  Degree,         .            .  141 — 150 

Problem  of  the  Lights,            .           • ,            .            .            .  150—151 
Equation  of  the  Second  Degree  with  Two  Unknown  Quantities,  151 — 155 

Extraction  of  the  Square  Root  of  the  Binomial  a±  \/^  ^^^ — 1^^ 


Formation  of  Poic 


CHAPTER  IV. 

and  Extract! o?i  of  Roots  of  any  degree 
whatever. 

ART1CLE3. 

159—161 

161—164 

164—168 

168—171 

171 

171—175 

175—177 

177—180 

180 

181 

182 

183—186 

186—189 

189 

190 

191 

192—194 

194 

195 

196—197 

199—201 

201—202 

203 

207—212 


Formation  of  Powers, 

Theory  of  Permutations  and  Combinations,     . 

Binomial  Tiieorem,      .... 

Consequences  of  Binomial  Theorem, 

Extraction  of  Roots  of  Numbers, 

Cube  Root,      ..... 

To  Extract  the  ?t' '  Root  of  a  Whole  Number, 

Extraction  of  Roots  by  Approximation, 

Cube  Root  of  Decimal  Fractions, 

Any  Root  of  a  Decimal  Fraction, 

Formation  of  Powers  and  Extraction  of  Roots, 

Of  Monomials — Of  Polynomials, 

Calculus  of  Radicals — Transformation  of  Radicals, 

Addition  and  Subtraction  of  Radicals, 

Multiplication  and  Division,     . 

Formation  of  Powers  and  Extraction  of  Roots, 

Different  Roots  of  Unity, 

Modifications  of  the  Rules  for  Radicals, 

Theory  of  Exponents, 

Multiphcation  of  Quantities  with  any  Exponent — Di 

Extraction  of  Roots,  .... 

Binomial  Theorem  for  any  Exponent — Applications, 

Degree  of  Approximation,       .  . 

Method  of  Indeterminate  Co-EfBcients — Recurring  Series, 

CHAPTER  V. 
Of  Progi^essions,  Continued  Fractions,  and  Logarithms. 

Progressions  by  Differences,    .....  213 

Last  Term,      .             .             .             .             .            ,             .  215 — 217 

Sum  of  the  Extremes — Sum  of  Q\e  Series,     .             .             .  217 

Ten  Problems — To  find  any  number  of  Means,          .             .  218 — 221 

Geometrical  Progression,         .....  221 


CONTENTS. 


ARTICLE? 

Last  Term — Sum  of  the  Series,          ....  222 — ^225 

Progressions  having  an  Infinite  Number  of  Terms,     .             .  225 — 227 

Ten  Problems— To  find  any  number  of  Means,           ,             .  228 

Solution  of  Four  Principal  Problems,  ....  229 — 230 

Continued  Fractions,  ......     231 237 

Exponential  Quantities,        <  .            .            .            ,            .  238 

Theory  of  Logarithms,             .....  239 — 241 

Multiplication  and  Division,     .....     241 243 

Formation  of  Powers  and  Extraction  of  Roots,           ,            .  243 — ^245 

General  Properties,      ......     24.5 249 

Logarithmic  and  Exponential  Series— Modulus—Transfor- 
mation of  Series,  .....     249 256 

CHAPTER  VI. 

General  Theoj-y  of  Equations. 

General  Properties  of  Equations,        ....  256—264 

Remarks  on  the  Greatest  Common  Divisor,     .  .  .     264 271 

Transformation  of  Equations,  ....     271 274 

Remarks  on  Transformations — Derived  Polynomials,              .  274 — 279 

Elimination,     .......     279 ^283 

Equal  Roots, 283—287 

CHAPTER  VII. 

Resolution  of  Numerical  Equations. 

General  Principles — First  Principle — Second  Principle,          .  287—290 

Limits  of  the  Roots  of  Equations,       ....     290 ^293 

Ordinary  Limit  of  Positive  Roots,       ....     293 294 

Newton's  Method  for  Smallest  Limits,  .  .  .     294 297 

Consequences  Deduced,  .....     297 302 

Descartes'  Rule,  ......     302 305 

Commensurable  Roots  of  Numerical  Equations,         .             .  305—307 

Of  Real  and  Incommensurable  Roots,  .  .  .     307 309 

Newton's  Methods  of  Approximation,             .            .            .  309—311 


ALGEBRA 


CHAPTER  I. 
Preliminary  Definitions  and  Remarks, 

1.  Quantity  is  a  general  term  embracing  every  thing  which 
admits  of  increase  or  diminution. 

2.  Mathematics  is  the  science  of  quantity. 

3.  Algebra  is  that  branch  of  mathematics  in  which  the  quanti- 
ties considered  are  represented  by  letters, and  the  operations  to  be 
performed  upon  them  are  indicated  by  signs. 

4.  The  sign  +,  is  called  flus ;  and  indicates  the  addition  of  two 
or  more  quantities.  Thus,  9+5  is  read,  9  plus  5,  or  9  augmented 
by  5. 

In  like  manner,  a+&  is  read,  a  plus  h  ;  and  denotes  that  the  quan- 
tity represented  by  a  is  to  be  added  to  the  quantity  represented 
hyh. 

5.  The  sign  — ,  is  called  minus ;  and  indicates  that  one  quantity 
is  to  be  subtracted  from  another.  Thus,  9— 5  is  read,  9  minus  5, 
or  9  diminished  by  5. 

In  like  manner,  a— &,  is  read,  a  minus  b,  or  a  diminished  by  b. 

6.  The  sign  x>  is  called  the  sign  of  multiplication;  and  when 
placed  between  two  quantities,  it  denotes  that  they  are  to  be  multi- 
plied  together.  The  multiplication  of  two  quantities  is  also  fre- 
quently  indicated  by  simply  placing  a  point  between  them.     Thus, 


10  ALGEBRA. 

33x25,  or  36.25,  is  read,  36  multiplied  by  25,  or  the  product  of 
36  by  25. 

7.  The  multiplication  of  quantities,  which  are  represented  by  let- 
ters, is  indicated  by  simply  writing  them  one  after  the  other,  without 
interposing  any  sign. 

Thus, a6  signifies  the  same  thing  as  axh,  or  as  a.& ;  and  ahc 
the  same  as  aX^Xc,  or  as  a.h.c.  It  is  plain  that  the  notation 
db,  or  ahc,  which  is  more  simple  than  axh,  or  aXbXc,  cannot  be 
employed  when  the  quantities  are  represented  by  figures.  For 
example,  if  it  were  required  to  express  the  product  of  5  by  6,  and 
we  were  to  write  5  6,  the  notation  would  confound  the  product  with 
the  number  56. 

8.  In  the  product  of  several  letters,  as  abc,  the  single  letters,  a,  b 
and  c,  are  called /actor-s  of  the  product.  Thus,  in  the  product  ab, 
there  are  two  factors,  a  and  b  ;  in  the  product  acd,  there  are  three, 
a,  c  and  d 

9.  There  are  three  signs  used  to  denote  division.     Thus, 

a-^b  denotes  that  a  is  to  be  divided  by  b, 

—       denotes  that  a  is  to  be  divided  by  b, 

a\b      denotes  that  a  is  to  be  divided  by  b. 
r 

10.  The  sign  =,  is  called  the  sign  of  equality,  and  is  read,  is 
equal  to.  When  placed  between  two  quantities,  it  denotes  that  they 
are  equal  to  each  other.  Thus,  9  — 5=4  :  that  is,  9  minus  5  is 
equal  to  4  :  Also,  a-{-b=c,  denotes  that  the  sum  of  the  quantities 
a  and  b  is  equal  to  c.  "^ 

11.  The  sign  >,  is  called  the  sign  of  inequality,  and  is  used  to 
express  that  one  quantity  is  greater  or  less  than  another. 

Thus,  ayb  is  read,  a  greater  than  b;  and  a<6  is  read,  a  less 
than  b  ;  that  is,  the  opening  of  the  sign  is  turned  towards  the  greater 
quantity. 

12.  If  a  quantity  is  added  to  itself  several  times,  as  a+a+a+a 
+a,  we  '  —orally  write  it  but  once,  and  then  nlaco  a  nnmber  before 


DEFIMTIONS  A>D  REMARKS.  11 

it  to  express  how  many  times  it  is  taken.     Thus, 
a-^a-\-a-\-a-{-a=5a. 

The  number  5  is  called  the  co-efficieni  of  a,  and  denotes  that  a  is 
taken  5  times. 

Hence,  a  co-efficient  is  a  number  prefixed  to  a  quantity,  denotmg 
the  number  of  times  which  the  quantity  is  taken;  and  it  also  indi- 
cates the  number  of  times  plus  one,  that  the  quantity  is  added  to 
itself.  When  no  co-efRcient  is  written,  the  co-efficient  1  is  always 
understood. 

13.  If  a  quantity  be  multiplied  continually  by  itself,  as  ax^Xa 
XaXfl)  we  generally  express  the  product  by  writing  the  letter 
once,  and  placing  a  number  to  the  right  of,  and  a  little  above  it :  thus, 

aX«X«X«Xa=«^ 

The  number  5  is  called  the  exponent  of  a,  and  denotes  the  number 
of  times  which  a  enters  into  the  product  as  a  factor. 

Hence,  the  exponent  of  a  quantity  shows  how  inany  times  the 
quantity  is  a  factor ;  and  it  also  indicates  the  number  of  times,  phis 
one,  that  the  quantity  is  to  be  multiplied  by  itself.  When  no  expo- 
nent is  written,  the  exponent  1  is  always  understood. 

14.  The  product  resulting  from  the  multiplication  of  a  quantity 
by  itself  any  number  of  times,  is  calledthe  power  of  that  quantity  j 
acd  the  exponent,  which  always  exceeds  by  onethe  number  of  mul- 
tiplications to  be  made,  denotes  the  degree  of  the  power.  Thus,  a' 
is  the  fifth  power  of  a.  The  exponent  5  denotes  the  degree  of  the 
power ;  and  the  power  itself  is  formed  by  multiplying  a  four  times 
by  itself. 

15.  In  order  to  show  the  importance  of  the  exponent  in  algebra, 
suppose  that  we  wish  to  express  that  a  number  a  is  to  be  multiplied 
three  times  by  itself,  that  this  product  is  to  be  multiplied  three  times 
by  h,  and  that  this  new  product  is  to  be  multiplied  twice  by  c,  we 
would  write  simply     a^  IP  (?. 

If,  then,  we  wish  to  expess  that  this  last  result  is  to  be  added  to 
itself  six  times,  or  is  to  be  multiplied  by  7,  we  would  write,  7a'6V. 


12 


ALGEBRA. 


This  gives  an  idea  of  the  brevity  of  algebraic  language. 

16.  The  root  of  a  quantity,  is  a  quantity  which  being  multiplied 
by  itself  a  certain  number  of  times  will  produce  the  given  quantity. 

The  sign  y/  ,  is  called  the  radical  sign,  and  when  prefixed  to 
a  quantity,  indicates  that  its  root  is  to  be  extracted.     Thus, 

Vfl     or  simply  ■v^a  denotes  the  square  root  of  a. 

"^  a    denotes  the  cube  root  of  a. 

^ a,   denotes  the  fourth  root  of  a. 
The  number  placed  over  the  radical  sign  is  called  the  index  of  the 
root.     Thus,  2  is  the  index  of  the  square  root,   3  of  the  cube  root, 
4  of  the  fourth  root,  &c. 

17.  Every  quantity  written  in  algebraic  language  ;  that  is,  with 
the  aid  of  letters  and  signs,  is  called  an  algebraic  quantity,  or  the 
algebraic  expression  of  a  quantity.     Thus, 

C  is  the  algebraic  expression  of  three  times  the 

i  number  a ; 

C  is   the  algebraic  expression  of  five   times  the 

(  square  of  a  ; 

c  is  the  algebraic  expression  of  seven  times  the 

i  product  of  the  cube  of  a  by  the  square  of  b  ; 

(     is  the  algebraic  expression  of  the  difference  be- 
3a  — 5o  <  ,  „        .         , 

(      tween  three  times  a  and  five  tunes  b ; 

^    is  the  algebraic  expression  of  twice  the  square 

2a^—Sab+4:b^7      of  a,  diminished  by  three  times  the  product  of  « 

(      by  b,  augmented  by  four  times  the  square  of  b. 

18.  When  an  algebraic  quantity  is  not  connected  with  any  other 
by  the  sign  of  addition  or  subtraction,  it  is  called  a  monomial,  or  a 
quantity  composed  of  a  single  term,  or  simply,  a  term. 

Thus,         3a,  5a-,  laW,  are  monomials,  or  single  terms. 

19.  An  algebraic  expression  composed  of  two  or  more  parts, 
separated  by  the  sign  +  or  — ,  is  called  a  polynomial,  or  quantity 
involving  two  or  more  terms. 


DEFINITIONS  AND  REMARKS.  13 

For  example,  3a  — 5 J  and  2a-—2cb-{-'il/  are  polynominls. 

20.  A  polynomial  composed  of  two  terms,  is  called  a  hinomial ; 
and  a  polynomial  of  three  terms  is  called  a  trinomial. 

21.  The  numerical  value  of  an  algebraic  expression, is  the  number 
which  would  be  obtained  by  giving  particular  values  to  the  letters 
which  enter  it,  and  performing  the  arithmetical  operations  indicated. 
This  numerical  value  evidently  depends  upon  the  particular  values 
attributed  to  the  letters,  and  will  generally  vary  with  them. 

For  example,  the  numerical  value  of  2a^=54  when  we  make 
a=3  ;  for,  the  cube  of  3=-27,  and  2x27=54. 

The  numerical  value  of  the  same  expression  is  250  when  we 
make  a=5;  for,  53=I25,and  2x125=250. 

22.  We  have  said,  that  the  numerical  value  of  an  algebraic  ex- 
pression generally  varies  with  the  values  of  the  letters  which  enter 
it :  it  does  not,  however,  always  do  so.  Thus,  in  the  expression 
a  —  h,  so  long  as  a  and  b  increase  by  the  same  number,  the  value 
of  the  expression  will  not  be  changed. 

For  example,  make  a=7  and  S=4 :  there  results  a  — Z»=3. 
Now  make    a=7+5=12,  and  ^=4  +  5=9,    and  there  results 
a  — ^=12  — 9=3,  as  before. 

23.  The  numerical  value  of  a  polynomial  is  not  affected  by 
changing  the  order  of  its  terms,  provided  the  signs  of  all  the  terms 
be  preserved.  For  example,  the  polynomial  4a''  — 3a-i+5ac'^= 
5af'-— 3a''^o+4a^=— 3a-Z'+5ac^+4a^.  This  is  evident,  from  the 
nature  of  arithmetical  addition  and  subtraction. 

24.  Of  the  different  terms  which  compose  a  polynomial,  some 
are  preceded  by  the  sign  +,  and  the  others  by  the  sign  — .  The 
first  are  called  additive  terms,  the  others,  suhtractive  terma. 

The  first  term  of  a  polynomial  is  commonly  not  preceded  by  any 
sign,  but  then,  it  is  understood  to  be  afTected  with  the  sign   +. 

25.  Each  of  the  literal  factors  which  compose  a  term  is  called  a 
dimension  of  this  term  ;   and  the  degree,  of  a  term  is  the  i  umber  of 

2 


14  ALGEBRA. 

these  factors  or  dimensions.     Thus, 

3a  is  a  term  of  one  dimension,  or  of  the  first  degree. 

hah  is  a  term  of  two  dimensions,  or  of  the  second  degree. 

la^hr=laaabcc  is  of  six  dimensions,  or  of  the  sixth  degree. 
In  genei-al,  the  degree,  or  the  number  of  dimensions  of  a  term,  is 
estimated  by  taking  the  stitn  of  the  exponents  of  the  letters  ivhich  enter 
this  term.     For  example,  the  term  Sa^bcd^  is  of  the  seventh  degree, 
since  the  sum  of  the  exponents,  2+1  +  1+3=7. 

26,  A  polynomial  is  said  to  be  homogeneous,  when  all  its  terms 
are  of  the  same  degree.     The  polynomial 

3a— 2&+C         is  of  the  first  degree  and  homogeneous. 
—  ^ab-\-¥  is  of  the  second  degree  and  homogeneous. 

5a^c— 4c''+2c^(Z  is  of  the  third  degree  and  homogeneous. 
8a^— 4aS+c       is  not  homogeneous. 

27,  A  vinculum  or  bar ,  or  a  parenthesis  (  ),  is  used  to 

express  that  all  the  terms  of  a  polynomial  are  to  be  considered  to- 


gether.  Thus,  a-\-b-\-cxb,  or  (a+Jxc)xJ  denotes  that  the 
trinomial  a-\-h-\-c  is  to  be  multiplied  by  b  ;  also  a-{-b-\-cxc+d-{-f 
or  (a+J  +  c)x(c+(^+y)  denotes  that  the  trinomial  a-\-b-\-c  is  to 
be  multiplied  by  the  trinomial  c-\-d-\-f 

When  the  parenthesis  is  used,  the  sign  of  multiplication  is  usually 
omitted.     Thus  {a-\-b-\-c)xb  is  the  same  as  {a-\-b-\-c)  b. 

The  bar  is  also  sometimes  placed  vertically.     Thus, 


+a 
+^ 
+  c 


is  the  same  as  {a-\-b-\-c)  x   or  a+b  +  cXx 


28.  The  terms  of  a  polynomial  which  are  composed  of  the  same 
letters,  the  same  letters  in  each  being  affected  with  like  exponents, 
are  called  similar  term^. 

Thus,  in  the  polynomial  lah  +  Sab—^taW+oa^l^,  the  terms  lab 
and  Sab,  are  similar ;  and  so  also  are  the  terms— 40^^  and  5a'Zr', 
the  letters  and  exponents  in  each  being  the  same.     But  in  the  bino- 


DEFINITIONS  AND  REMARKS.  15 

mial  8a^b-{-7aP,  the  terms  are  not  similar  ;  for,  although  they  are 
composed  of  the  same  letters,  yet  the  same  letters  are  not  affected 
with  like  exponents, 

29,  When  a  polynomial  contains  several  shnilar  terms  it  may 
often  be  reduced  to  a  simpler  form. 

Take  the  polynomial  4a^Z>— 3a-c+7a^c  — 2a-^. 
It  may  be  written  (Art.  23),  4a"b—2a^b-{-7a^c  —  3a'c, 
But  4:a^b—2a^b  reduces  to  2arb,  and  7 arc— Sac  to  4a^c, 
Hence,  4a^J — Sa'c + 7 ah — 2a'b=  2a^ + Aa^., 
When  we  have  a  polynomial  with  similar  terms,  of  the  form 
+  2a'br  -  4.a^bc'-\-6a'b(r—8a^bc^+  lla'bc^ 
Find  the  sum  of  the  additive  and  subtractive  terms  separately,  and 
take  their  difference  :  thus, 

Additive  terms,  Subtractive  terms, 

+  2a''b(?  -   4a^c^ 

+  Ga^c"  -   Qa^b(? 


-^\Wb(?  Sum  -\2a%e' 


Sum       +19a='Ac2 

Hence,  the  given  polynomial  reduces  to 

\^a''bc^-\2d'b(?=7a^b<?. 

It  may  happen  that  the  sum  of  the  subtractive  terms  exceeds  the 
sum  of  the  additive  terms.  In  that  case,  subtract  the  positive  co- 
efficient  from  the  negative,  and  prefix  the  minus  sign  to  the 
remainder. 

Thus,  in  the  polynomial,  Za%+2a^b—ba%—3a^b,  in  which  the 
sum  of  the  additive  terms  is  ba%  and  the  sum  of  the  subtractive 
terms    —Sa'Z',  we  say  that  the  polynomial  reduces  to    —Za^b, 

For,   since   —  8a=3    is    equal  to    ~ba%—3a%   we  shall    have, 
ba%  —  Qa^b= hdb — hdrb — 3a%=  —  2,cC-b. 

Hence,  for  the  reduction  of  the  similar  terms  of  a  polynomial  we 
have  the  following 


16 


ALGEBRA. 


RULE. 

I.  Form  a  single  addilive  term  of  all  the  terms  preceded  by  the  sign 
plus :  this  is  done  by  adding  together  the  co-efficienis  of  those  terms, 
and  annexing  to  their  sum  the  literal  part, 

II.  Form,  in  the  same  manner,  a  single  subtractive  term, 

III.  Subtract  the  less  sum  from  the  greater,  and  prefix  to  the 
result  the  sign  of  the  greater. 

Remark. — It  should  be  observed  that  the  reduction  affects  only 
the  co-efficients,  and  not  the  exponents. 

EXAMPLES. 

1.  Reduce  the  polynomial  Aa^b—Sa^b  —  da^b-\-\\a^b  to  its  sim- 
plest form.  Ans.     —2a^b. 

2.  Reduce  the  polynomial  7abc'^—abc^  —  7abc'—8abc^-\-6abc^  to 
its  simplest  form.  A71S,      — 2abc^, 

3.  Reduce  the  polynomial  9cP  —  Sac'^+15cP  +  8ca  +  9ac'—24.cP 
to  its  simplest  form.  Ans,     ac^-\-8ca. 

The  reduction  of  similar  terms  is  an  operation  peculiar  to  algebra. 
Such  reductions  are  constantly  made  in  Algebraic  Addition,  Sub- 
traction,  Multiplication,  and  Division, 

30.  It  has  been  remarked  in  Definition  3,  that  the  quantities  con- 
sidered in  algebra  are  represented  by  letters,  and  the  operations  to 
be  performed  upon  them,  are  indicated  by  signs.  The  letters  and 
signs  are  used  to  abridge  and  generalize  the  reasoning  required  in  the 
resolution  of  questions. 

31.  There  are  two  kinc^  of  questions,  viz.  theorems  and  problems. 
If  it  is  required  to  demonstrate  the  existence  of  certain  properties 
relating  to  quantities,  the  question  is  called  a  theorem ;  but  if  it  is 
proposed  to  determine  certain  quantities  from  the  knowledge  of 
others,  which  have  with  the  first  known  relations,  the  question  is 
culled  a  ptroblem. 

The  given  or  known  quantities  are  generally  represented  by  the 
first  letters  of  the  alphabet,  a,  h,  c,  d,  &c.  and  the  unknown  or  re- 


DEFINITIONS  AND  REMARKS.  17 

quired  quantities  by  the  last  letters,  x,  y,  z,  &c. 

32.  The  following  question  will  tend  to  show  the  utility  of  the  alge- 
braic analysis,  and  to  explain  the  manner  in  which  it  abridges  and 
generalizes  the  reasoning  required  in  the  resolution  of  questions. 

Question. 

The  sum  of  two  numbers  is  67,  and  their  difference  19 ;  what  are 
the  two  numbers  1 

Solution. 

We  will  begin  by  establishing,  with  the  aid  of  the  conventional 
signs,  a  connexion  between  the  given  and  unknown  numbers  of  the 
question.  If  the  least  of  the  two  required  numbers  was  known,  we 
would  have  the  greater  by  adding  19  to  it.  This  being  the  case, 
denote  the  least  number  by  x :  the  greater  may  then  be  designated 
by  a;+19 :  hence  their  sum  is  x-\-x-\-l9,  or  2.r+19. 

But  from  the  enunciation,  this  sum  is  to  be  equal  to  67.  There- 
fore we  have  the  equality  or  equation 

2a:+19=67. 

Now,  if  2x  augmented  by  19,  gives  67,  2x  alone  is  equal  to  67 
minus  19,  or  2a;=67  — 19,  or  performing  the  subtraction,  2xz=:48. 

Hence  x  is  equal  to  the  half  of  48,  that  is. 

The  least  number  being  24,  the  greater  is 
x+19=24  +  19^43. 
And  indeed, we  have  43+24=67,  and  43—24=19, 

Table  of  the  Algebraic  Operations. 
Let  X   be  the  least  number. 

x+19     will  be  the  greater. 
Hence,     2x+19=67,and2a;=67— 19  ;  therefore  x=Y= 24  and 
consequently  x+19=24+19=43. 

And  indeed,  43+24=67,43-24=19. 

Another  Solution. 

Let  X  represent  the  greater  number, 
2=. 


^®  ALGEBRA. 

•r 

a:— 19    will  represent  the  least. 

Hence,  2a;- 19=67,  whence  2a;=67  +  19  ; 
therefore,  a;=  5-6  =  43 

and  consequently,  a;  — 19=43  — 19=24. 

From  this  we  see  how  we  might,  with  the  aid  of  algebraic  signs, 
write  down  in  a  very  small  space,  the  whole  course  of  reasoning 
which  it  would  be  necessary  to  follow  in  the  resolution  of  a  prob- 
lem,  and  which,  if  written  in  common  language,  would  often  require 
several  pages. 

General  Solution  of  this  Problem, 

The  sum  of  two  numbers  is  a,  their  difference  is  h.  What  are 
the  two  numbers  ? 

Let  X  be  the  least  number, 

x-{-h  will  represent  the  greater. 
Hence,2a'-f  i=cf,  whence  2x=:a  —  b, 

.1        r.  a  —  b      a      b 

tlierefore,  x= = 

2        2     2 

and  consequently, a:+5= \-b=—-\- 

2        2  2        2 

As  the  form  of  these  two  results  is  independent  of  any  particular 
value  attributed  to  the  letters  a  and  b,  it  follows  that,  knoioing  the 
sum  and  difference  of  two  numbers,  we  tvill  obtain  the  greater  by  add- 
ing  the  half  difference  to  the  half  sum,  and  the  less,  by  subtracting  the 
half  difference  from  half  the  suin. 

Thus,  when  the  given  sum  is  237,  and  the  difference  99, 

,      .  237      99,        237  +  99       336      ,^^ 

the  greater  IS f- —  or ■ = =168; 

2         2  2  2 

,    ,     ,  2.37      99,        138      ^ 

aiwl  the  least —  or —69, 

2         2  2 

And  indeed,  168  +  69=237,  and  168  —  69=99. 

From  the  preceding  question  we  perceive  the  utility  of  repre. 

sontiog   the    given  quantitioa  of  a   problem    by  letters.     As   tho 


DEFINITIONS  AND  REMARKS. 


19 


arithmetical  operations  can  only  be  indicated  upon  these  letters, 
the  result  obtained,  points  out  the  operations  which  are  to  be  per- 
formed  upon  the  known  quantities,  in  order  to  obtain  the  values  of 
those  required  by  the  question. 

The  expressions  — + —  and  -j- obtained  in  this  prob- 
lem, are  called  formulas,  because  they  may  be  regarded  as  com- 
prehending  the  solutions  of  all  questions  of  the  same  nature,  the 
enunciations  of  which  differ  only  in  the  numerical  values  of  the 
given  quantities.  Hence,  a  formula  is  the  algebraic  enunciation  of 
a  general  rule. 

From  the  preceding  explanations,  we  see  that  Algebra  may  be 
regarded  as  a  kind  of  language,  composed  of  a  series  of  signs,  by 
the  aid  of  which  we  can  follow  with  more  facility  the  train  of  ideas 
in  the  course  of  reasoning,  which  we  are  obliged  to  pursue,  either  to 
demonstrate  the  existence  of  a  property,  or  to  obtain  the  solution  of 
a  problem. 

ADDITION. 

33.  Addition,  in  Algebra,  consists  in  finding  the  simplest  equiva- 
lent expression  for  several  algebraic  quantities,  connected  together 
by  the  sign  plus  or  minus.  Such  equivalent  expression  is  called 
their  sum. 

{3a 

The  result  of  the  addition  is 3a+5^-f-2c 

an  expression  which  cannot  be  reduced  to  a  more  simple  form. 

r  Aa"l? 

Again,  add  together  the  monomials  )  2a'1P 

\  lab"' 

The  result,  after  reducing  (Art.  29),  is  .  .  l^d-b" 


20 


ALGEBRA. 


Let  it  be  required  to  find  the  sum 
of  the  expressions. 


3a2_4a5 

2ah-bl^ 


Their  sum,  after  reducing  (Art.  29),  is  .  .  5a^— 5a&— 4^ 

35.  As  a  course  of  reasoning  similar  to  the  above  would  apply  to 
all  polynomials,  we  deduce  for  the  addition  of  algebraic  quantities 
the  following  general 

RULE. 

I.  Write  down  the  quantities  to  ie  added  so  that  the  similar  terms 
shall  fall  under  each  other,  and  give  to  each  term  its  proper  sign. 

II.  Reduce  the  similar  terms,  and  annex  to  the  results,  those  terms 
which  cannot  he  reduced,  giving  to  each  term  its  respective  sign. 

EXAMPLES. 

1.    Add    together  the    polynomials,    ^a'—2lr—4:al,  ba^—lr-{-2ah, 


The  term  Sa^  being  similar  to  ba\  we  C  S}^^—4ab—2l^ 
write  8a^  for  the  result  of  the  reduction  j  5a^-{-2ab—  P 
of  these  two  terms,  at  the   same  time  1  -^3ah— 2^—36^ 


•\ 


slightly  crossing  them,  as  in  the  first  term.   (^      8a^+   ab—blf—Sc' 

Passing  then  to  the  term  —  4a&,  which  is  similar  to  +2ab  and 
+  Sab,  the  three  reduce  to  +ab,  which  is  placed  after  Sa^  and  the 
terms  crossed  like  the  first  term.  Passing  then  to  the  terms  involving 
^,  we  find  their  sum  to  be  — 5^,  after  which  we  write  —  3c^ 

The  marks  are  drawn  across  the  terms,  that  none  of  them  may 
be  overlooked  and  omitted. 


Sum. 


(2). 

(3). 

7x  +3a5+2c 

8v/^+  bc-2ahc 

-3x  —  3aJ— 5c 

—    V^  —  9ic+6abc 

5a;   —9a  J— 9c 

—  5vT"+   hc-j-  ahc 

9x   —9a*— 12c 

2^^-lbc  +  babc 

SUBTRACTION. 


21 


4.  Add  together  the  polynomials  da-b  +  6cx-i-9bc\  Icx  —  Sa'b-^ 
'^a    and  —\hcx—U&\'2cC-h. 

Arts,     "^a  —a^b—2cx. 

5.  Add  together  the  polynomials    ^x  -\-ax—ab,   ab—  ^x  +ry, 
ax-\-xy—^<xb,   ^x  +  V^  — a;  and  xy-\-xy-\-ax. 

Am.     2  v/ x  +  3aa; — ^ab + ^xy — x. 
'  6;  Add  together  the  polynomials  lbaxy+fjbc^-\-^af",  ^ap+  ^xy 
—  V2xay,  —  bhc^-{-  ^liy —?>axy,  and  -2  ^ay—  "^x  —Qap. 

Ans,      ^/^_^/^-^^7. 
7.   Add    together   the   polynomials   7a'6  — 3aic  — 8Zrc  — 9c'+ca^, 
8ak-5a-5+3c3-45-c+cd2  and  ^a^b-Sc'+^Wc—ZcP. 

Ans.     Qd'b  -^babc  —  Z¥c  —  1 4c '  +  2c(?  —  3(Z'. 

SUBTRACTION. 

36.  Subtraction,  in  algebra,  consists  in  finding  the  simplest  ex- 
pression for  the  difTerence  between  two  algebraic  quantities. 

The    result  obtained  by  subtracting   4^   from   5a   is  expressed 
by     5a  — 4&. 

In  like  manner,  the  difference  between  '{(Cb  and  Aa^b  is  expressed 
by  la^b-A.a^b^2a^b. 

Let  it  be  required  to  subtract  from       .  .  .4a 

the  binomial        .  .  .  .      ■    .  .2b  —  Sc 


In  the  first  place,  the  result  may  be  written  thus,  4a  — (2i— 3c) 
by  placing  the  quantity  to  be  subtracted  within  the  parenthesis,  and 
writuig  it  after  the  other  quantity  with  the  sign  — .  But  the  ques- 
tion frequently  requires  the  difference  to  be  expressed  by  a  single 
polynomial ;  and  it  is  in  this  that  algebraic  subtraction  principally 
consists. 

To  accomplish  this  object,  we  will  observe,  that  if  a,  b,  c,  were 
given  numerically,  the  subtraction  indicated  by  23— 3c,  could  be 
performed,  and  we  might  then  substract  this  result  from  4a ;  but  as 


22  ALGEBRA. 

this  subtraction  cannot  be  effected  in  the  actual  condition  of  the 
quantities,  2b  is  subtracted  from  4a,  which  gives  4a— 2i ;  but  in  sub- 
tracting  the  number  of  units  contained  in  2b,  the  number  taken 
away  is  too  great  by  the  number  of  units  contained  in  3c,  and  the 
result  is  therefore  too  small  by  the  same  quantity  ;  this  result  must 
therefore  be  corrected  by  adding  3c  to  it.  Hence,  there  results 
from  the  proposed  subtraction  4a— 2J  +  3c. 
4a 
2b-Sc 


4a  — 25+ 3c 


Again,  from           .....     8a"—2ab 
subtract dci'—Aab+Shc  —  P. 

The  difference  is  expressed  by  8ar—2ab—(5ar—4:ab+Sbc—lf^) 
which  is  equal  to        .         .  .      8a^— 2a5— 5a-— 3ic  +  4a5+i^. 

or  by  reducing,  equal  to  .  .  .         .     Sa^-{-2ab—Sbc-\-l^. 

The  reduction  is  made  by  observing,  that  to  subtract  5a^— 4a5 
-^Sbc  —  I^,  is  to  subtract  the  difference  between  the  sum  of  the  ad- 
ditive terms  5a^-{-Bbc,  and  the  sum  of  the  substractive  terms  4ai+^. 
We  can  then  first  subtract  5a^+3bc,  which  gives  8a^— 2ai— 5a^ 

—  3bc;  and  as  this  result  is  necessarily  too  small  by  4ab+P,  this 
last  quantity  must  be  added  to  it,  and   it  becomes  8a^— 2aZi  — 5a^ 

—  3bc  +  4:ab-\-P  ;  and  finally,  after  reducing,  3a^-\-2ab—3bc+P. 

37.  Hence,  for  the  subtraction  of  algebraic  quantities,  we  have  the 
following  general 

RULE. 

I.  Write  the  quantity  to  be  subtracted  under  that  from  lohich  it  is 
to  be  taken,  placing  the  similar  terms,  if  there  are  any,  under  each 
other. 

II.  Change  the  signs  of  all  the  terms  of  the  -polynomial  to  be  sub- 
tracted,  or  conceive  them  to  be  changed,  and  then  reduce  the  polynomial 
result  to  its  simplest  form. 


SUBTRACTION. 


23 


From 
Take 
Remainder 


From 
Take.     . 
Remainder 

From  .  . 
Take  .  . 
Remainder 


^d 


6ac  —  bab+<^ 
■  Sac—Sab-\-lc 

Sac—8ab-\-c'+'!c. 


6ac—5ah-\-c? 
Sac  +  Sab— 1c 
Sac  —  8ab-\-c^+lc. 
_(2). 

6\/2«y—  \r^-{-s¥ 

■SV2;ry-Vx+x+2U'.        byx-Sx'+S  +  bb- 


(3). 
Qyx—S3?-\-U 
yx—S    +  a 


(4). 
-ba^'—Aa^b+S^c 
-2a^+Sa^b-Sl^c 
7a^-la^b-\-llI^c. 


(5). 
4ab-   cd  +  Sa' 
5ab—4:cd+Sa-+5b' 


ab+Scd—blr 


7.   From  8abc  —  l2Pa  +  5cx—lxy,  take  Icx—xy  —  lSb^a. 

An^,     8abc-\-Pa  —  2cx—6ry. 

38.  By  the  rule  for  subtraction,  polynomials  may  be  subjected  to 
certain  transformations. 

For  example  .  .     6a^-Sab+2if'-2bc, 

becomes  .  .  .     6a''—{Sab—2P  +  2bc). 

In  like  manner  .  .      '!a'-8a'b-Alr'c+6P, 

becomes  .  .  .     '7a'*-{8a^+APc-6P)  ; 

or,  again,  .         .  .     la''-8a^-{4Pc-6by 

These  transformations  consist  in  decomposing  a  polynomial  into 
two  parts,  separated  from  each  other  by  the  sign  — :  they  are  very 
useful  in  algebra, 

39.  Remark.  — From  what  has  been  shown  in  addition  and  sub- 
traction,  we  deduce  the  following  principles. 

1st.  In  algebra,  the  words  add  and  sum  do  not  always,  as  in 
arithmetic,  convey  the  idea  of  augmentation ;  for  a—b,  which 
results  from  the  addition  of  —b  to  a,  is  properly  speaking,  a  dif- 
ference between  the  number  of  units  expressed  by  a,  and  the  num- 
ber of  units  expressed  by  b.    Consequently,  this  result  is  less  than  a. 


24  ALGEBRA. 

To  distinguish  this  sum  from  an  arithmetical  sum,  it  is  called  tho 
algebraic  sum. 

Thus,  the  polynomial  2rt~— 3fl-Z»+3//c  is  an  algebraic  sum,  so 
long  as  it  is  considered  as  the  result  of  the  union  of  the  monomials 
2a^,  —  Srt^^,  -\-2,}rc,  with  their  respective  signs  ;  and,  in  its  frofer  ac- 
ceptatwn,  it  is  the  arithmetical  difference  between  the  sum  of  the 
units  contained  in  the  additive  terms,  and  the  sum  of  the  units  con- 
tained in  the  subtractive  terms. 

It  follows  from  this  that  an  algebraic  sum  may,  in  the  numericol 
applications,  be  reduced  to  a  negative  number,  or  a  number  affected 
with  the  sign  — . 

2d.  The  words  siiUraction  and  difference  do  not  always  convey 
the  idea  of  diminution,  for  the  difference  between  +a  and  —b  being' 
a  +  b,  exceeds  a.  This  result  is  an  algebraic  difference,  and  can  be 
put  under  the  form  of  a  — (  — Z*). 

MULtlPLICATION. 

40.  Algebraic  multiplication  has  the  same  object  as  arithmetical, 
viz.  to  repeat  the  multiplicand  as  many  times  as  there  are  units  in 
the  multiplier. 

It  is  generally  proved,  in  arithmetical  treaties,  that  the  product  of 
two  or  moi-e  numbers  is  the  same,  in  whatever  order  the  multiplii  a- 
tion  is  performed  ;  we  will,  therefore,  consider  this  principle  de- 
monstrated. 

This  being  ndmitted,  we  v/ill  first  consider  the  case  in  which  it  is 
required  to  ninltiply  one  monomial  by  another. 

The  expression  for  the  ])roduct  of      .    la^V^  by  A,a-b 
may  at  once  be  written  thus        .         .     laVrx^ba^ 

But  this  may  be  simplified  by  observing  that,  from  the  preceding 
principles  and  the  signification  of  algebraic  symbols,  it  can  be 
written       .  .  .     7x^anaoahbb. 

Now,  as  t!ie  co-efficients  are  particular  numbers,  nothing  prrvonfs 
our  forming  a  single  number  from  them  by  multiplying  them 
together,  which  gives  28  for  thf*  co-efficient  of  the  product.     As  to 


MULTIPLICATION.  25 

the  letters,  the  product  aaaaa,  is  equivalent  to  a-,  and  the  product 
hbb,  to  IP  ;  therefore,  the  final  result  is    .     .     .    2QaW. 

Again,  let  us  multiply    ....     Vla^b'c^  by  Qd^JroP. 
The  product  is         V2x^aaaaabbbhbhccdd=9QaW(?cP. 

41.  Hence,  for  the  multiplication  of  monomials  we  have  the 
following 

RULE. 

I.  Multiply  the  co-efficients  together. 

II.  Write  after  this  product  all  the  letters  ivhich  are  cotnmon  to  the 
multiplicand  and  multiplier,  affecting  each  letter  ivith  an  exponent 
equal  to  the  sum  of  the  two  exponents  with  which  this  letter  is  affected 
in  the  two  factors. 

III.  If  a  letter  enters  into  but  one  of  the  factors,  write  it  in  tlw  pro- 
duct with  the  exponent  with  which  it  is  affected  in  the  factor. 

The  reason  for  the  rule  relative  to  the  co-efRcients  is  evident.  But 
in  order  to  understand  the  rule  for  the  exponents,  it  should  be  ob- 
served, that  in  general, a  quantity  a  is  found  as  many  times  a  factor 
in  the  product,  as  it  is  in  both  the  multiplicand  and  multiplier.  Now 
the  exponents  of  the  letters  denote  the  number  of  times  they  enter  as 
factors  (Art.  13.)  ;  hence  the  sum  of  the  two  exponents  of  the  same 
letter  denotes  the  number  of  times  it  is  a  factor  in  the  required 
product. 

From  the  above  rule,  it  follows  that, 

8a''bc'x'iabd?=   bQaWc'cP 
2laWdc  X  8abc''=16&a'Pc'd 
^abcx'!df=   28abcdf. 
Multiplv    .     Sa^b  12  a'x  Qxy  z  a-xy 

by    .'     .     2b  a^  12  x^y  ayH  Ixf 

6a!^lP  144rtV7/  Sxuy^z'  2a\v'y^ 


42.  We  will   now  proceed  to  the  multiplication  of  polynomials. 

Take  the  two  polynomials  a-{-b-}-c,  and  d+f  com.posed  entirely  of 

additive  terms ;  the   product  may    be    presented    under  the   form 

(a  +  S  +  c)  (d-{-f).      But   it    is  often   necessary   to  form  a  single 

3 


26 


ALGEBRA. 


polynomial  from  this  product,  and  it  is  in  this  that  the  multiplication 
of  two  polynomials  consists. 

Now  it  is  evident,  that  to  multiply   .     a+b-\-c 
by d+f 


ad-{-bd-\-cd 
+af+bf+cf 


ad+bd+cd+ a/+  bf+  tf. 

is  the  same  thing  as  taking  a-\-b-\-c  as  many  times  as  there  are  units 
in  d,  then  as  many  times  as  there  are  units  in^,  and  adding  the  two 
products  together.  But  to  multiply  a  +  ^-f  c  by  d,  is  to  take  each  of 
the  parts  of  the  multiplicand  d  times  and  add  together  the  partial 
products,  which  gives  ad-\-bd-\-cd.  In  like  manner,  to  multiply 
a-\-b-^c  by/,  is  to  take  each  of  the  parts  of  the  multiplicand,/times, 
and  add  together  the  partial  products. 

Hence,     (a  +  b+c)  {d-^f)T=ad-\-bd+cd+af-^bf+cf. 

Therefore,  in  order  to  multiply  together  two  polynomials  com- 
posed  entirely  of  additive  terms,  multiply  successively  each  term  of  the 
multiplicand  by  each  term  of  the  inultiplier,  and  add  together  all  the 
j)roducts. 

If  the  terms  are  affected  with  co-efficients  and  exponents,  observe 
the  rule  given  for  the  multiplication    of  monomials  (Art.  41). 


For  example,  multiply 

by  .          .         .         . 

.     2a+  5b 

The  product,  after  reducing, 

60""+   Qa'b-^2aV' 

+  \ba^b+2{)a¥+W 

becomes 
x+y 

.     6a'+23a^*+22a^'+5Z'3 

a;'+   xf +lax 
ax  +bax 

+^y+f 
x'*+xf+x'y+y' 

a3?+axY  +  'id'3? 

+  5ax«  +5flxy+35aV 
6aa;''+6aarj/«+42aV. 

MULTIPLICATION. 


27 


43.  In  order  to  explain  the  most  general  case,  we  will  first  re- 
mark, that  if  the  multiplicand  contains  additive  and  subtractive 
terms,  it  may  be  considered  as  expressing  the  difference  between  the 
number  of  units  indicated  by  the  sum  of  the  additive  terms,  and  the 
number  of  units  indicated  by  the  sum  of  the  subtractive  terms.  The 
same  reasoning  applies  to  the  multiplier  ;  whence  it  follows,  that  the 
general  case  may  be  reduced  to  the  multiplication  of  two  binomials, 
such  as  a  —  b  and  c—d',  a  denoting  the  sum  of  the  additive  terms, 
and  b  the  sum  of  the  subtractive  terms  of  the  multiplicand,  c  and  d 
expressing  similar  values  of  the  multiplier.  We  will  then  show  how 
the  multiplication  expressed  \)y{a  —  b)x{c  —  d)  can  be  effected, 

a  -b 

c  -d 

ac  —  bc 
—  ad+bd 


ac—bc  —ad-\-bd. 


Now,  to  multiply  a—b  by  c—d,  is  evidently  the  same  thing  as  to 
take  a  —  b  as  many  times  as  there  are  units  in  c,  and  then  diminish  this 
product  by  a—b,  taken  as  many  times  as  there  are  units  in  d;  or  to 
multiply  a—b  by  c,  and  subtract  from  this  product  that  of  a— J  by  d. 
But  to  multiply  a—bhyc,  is  to  take  a—b,c  times.  Now  if  we  mul- 
tiply  a  by  c  the  product  is  ac,  which  is  too  large  by  b  taken  c  times ; 
therefore  cimust  be  taken  from  it :  hence,  the  product  of  a— 5  by  c, 
is  ac  —  bc.  In  like  manner,  the  product  of  a  —  bhy  d,  is  ad  —  bd; 
and  as  we  have  just  seen  that  this  last  product  should  be  subtract- 
ed  from  the  preceding  ac—bc,  it  is  necessary  to  change  the  signs  of 
ad—bd,  and  write  it  under  ac  —  bc,  which  (Art.  37),  gives 
(a  —  b)  (c  —  d)=ac—bc  —  ad+bd. 

If  we  suppose  a  and  c  each  equal  to  0,  the  product  will  reduce 
to  +bd. 

44.  Hence,  for  the  multiplication  of  one  polynomial,  by  another 
we  have  the  following 


28  ALGEBRA. 

EULE. 

I.  Multiply  aU  the  terms  of  the  multrpllcand,  both  additive  and  sub. 
tractive,  by  each  additive  term  of  the  multiplier,  and  affect  the  partial 
products  with  the  same  signs  as  those  with  which  the  terms  of  the  nnil- 
tiplicand  are  affected ;  also  midtiply  all  the  tentis  of  the  multiplicand 
by  each  sidHractive  term  of  the  multiplier,  but  affect  the  partial  products 
with  signs'Conirnry  to  those  with  which  the  terms  of  the  multiplicand  are 
affected.     Then  reduce  the  pohnondal  result  to  its  simplestforrn. 

Take,  for  an  example,  the  two  polynomials  : 
4a:'  —  5a-b  —  8ab''-\-2P 
and  2a--Sab-4I/' 


8a'—  lOa'b—  WaW+Aa'b'' 

—  V2a'b  +lbaW+24:a~b^-Qab'' 

—  16a''^-  +  20a'Z-H32a  b'  —  8l/- 


8aP-22a'b—lla'l^+A.8a-b-  +  2Qab'  —  8lf, 
After  having  arranged  the  polynomials  one  under  the  other,  mul- 
tiply  each  term  of  the  first,  by  the  tenn  2a^  of  the  second  ;  this  gives 
the  polynomial  8a^—lQa^b  —  \Qa¥-^AaW,  the  signs  of  which  are 
the  same  as  those  of  the  multiplicand.  Passing  then  to  the  term 
?>ah  of  the  multiplier,  multiply  each  term  of  the  multiplicand  by  it, 
and  as  it  is  affected  with  the  sign  — ,  affect  each  product  with  a  sign 
contrary  to  that  of  the  corresponding  term  in  the  multiplicand  ;  this 
gives  —  12a*J4-15a^^+24a^5^— 6aJ*  for  a  product,  which  is  written 
under  the  first. 

The  same  operation  is  also  performed  with  the  term  AJr,  which  is 
also  subtractive;  this  gives,  —\Qa''V--^r20aW  +  ^2ab'  —  8b\  The 
product  is  then  reduced,  and  we  finally  obtain,  for  the  most  simple 
expression  of  the  product, 

8a'-22a'b-\laW-^^8a"F  +  2Qa¥-81/. 

The  rule  for  the  signs,  which  is  the  most  important  to  retain,  in  the 

multiplication  of  two  polynomials,  may  be  expressed  thus :    When  two 

terms  of  the  multiplicand  and  multiplier  are  affected  with  the  sa7ne 

sign,  the  corresponding  j)roduct  is  affected  with  the  sign  +,  and  when 


5IULTIPLICATION. 


29 


they  are  affected  ivith  contrary  signs,  the  product  is  affected  with  the 
sign  -. 

Again,  we  say  in  algebraic  language,  that  +  multiplied  by  +, 
or  —  multiplied  by  — ,  gives  +  ;  —  multiplied  by  +,  or  +  multi- 
plied by  — ,  gives  — .  But  this  last  enunciation,  which  does  not  in 
itself  offer  any  reasonable  direction,  should  only  be  considered  as  an 
abbreviation  of  the  preceding. 

This  is  not  the  only  case  in  which  algebraists,  for  the  sake  of 
brevity,  employ  incorrect  expressions,  but  which  have  the  advantage 
of  fixing  the  rules  in  the  memory. 

EXAMPLES. 


1. 

Multiply 

V2ax  by  3a. 

Am.     dda'x. 

2. 

Multiply 

U^-2y  by  2y. 

Ans.     8x'y- 

-Af. 

3. 

Multiply 

2x4-4^  by  2x—4:y. 

Ans.     AaP- 

ley. 

4. 

Multiply 

aP  J^x'y^xy'+y^  by  x-y. 

Ans.     X 

^-rf. 

5. 

Multiply 

x'+xy-itf  by  ^—xy-\-f. 

Ans.     x'^-\-x'y 

=+2/^ 

6. 

Multiply  ' 

2a=— 3ax+4ar^  by  ba^—^ax 

-2x^. 

7. 

Multiply 

Zx^—2xy-\-b  by  3?  +  2xy-[ 

3. 

8. 

Multiply  3af'+2a.-/+3/  ^y  2x'-3x' 

y+5f. 

9. 

.  red. 
1. 

2a'-5id+cf 
-5a2+4M-8c/. 

'rod 

-15a^  +  37a=M-29aV- 

2QU'd?-\-Ubcdf- 

-Srp. 

IC 

4a^&'-5aWc+8a'bc'-: 

^aV-labc^ 

2a&'-Aabc-2bc^  +, 

6\ 

r        8aW    —lOaWc -^2&aWc'-S4:aWc^ 
Prod.  red.       <j    -   4a'Pc''-lQa'b''c -{-V2a^  c*+  laWc^ 
[  +140^5  c^  +14a  ^»V-   3rtV    —lab  c\ 
45.   We  will  make  some  important  remarks  upon  algebraic  mul- 
tiplication. 

1st.  If  the  polynomials  proposed  to  be  multiplied  by  each  other 
are  homogeneous,  the  product  of  these  tuv  polynomials  ic ill  also  be  ho. 
3* 


30 


ALGEBRA. 


mogeneous.  This  is  an  evident  consequence  of  the  rules  relative  to  the 
letters  and  exponents  in  the  multiplication  of  monomials.  Moreover, 
the  degree  of  each  term  of  the  product  should  be  equal  to  the  sum  of 
the  degrees  of  any  two  terms  of  the  multiplier  and  multiplicand. 
Thus,  in  example  9,  all  the  terms  of  the  multiplicand  being  of  the 
second  degree,  as  well  as  those  of  the  multiplier,  all  the  terms  of  the 
product  are  of  the  fourth  degree.  In  example  10,  the  multiplicand 
being  of  the  fifth  degree,  and  the  multiplier  of  the  third,  the  product 
is  of  the  eighth  degree.  This  remark  serves  to  discover  any  errors 
in  the  calculations  with  respect  to  the  exponents.  Yox  example,  if 
it  is  found  that  in  one  of  the  terms  of  a  product  that  should  be  homo- 
geneous, the  sum  of  the  exponents  is  equal  to  7,  •vi\\\\e  in  all  the  others 
their  sum  is  8,  there  is  a  manifest  error  in  the  addition  of  the  expo- 
nents,  and  the  multiplication  of  the  two  terms  which  have  formed 
this  product  must  be  revised. 

2d.  When,  in  the  multiplication  of  two  polynomials,  the  product 
does  not  present  any  similar  terms  for  reduction,  the  total  number  of 
terms  in  the  product  is  equal  to  the  product  of  the  number  of  terms 
in  the  multiplicand,  multiplied  by  the  number  of  tenns  in  the  multi- 
plier. This  is  a  consequence  of  the  rule,  (Art.  44).  Thus,  when 
there  are  five  terms  in  the  multiplicand,  and  four  in  the  multiplier, 
there  are  5x4,  or  20,  in  the  product.  In  general  when  the  multi- 
plicand is  composed  of  m  terms,  and  the  multiplier  of  n  terms,  the 
product  contains  mXn  terms. 

3d.  When  some  of  the  terms  are  similar,  the  total  number  of 
terms  in  the  product,  when  reduced,  may  be  much  less.  But  we 
will  remark,  that  among  the  different  terms  of  the  product,  there  are 
some  that  cannot  be  reduced  with  any  others.  These  are,*"  1st. 
The  term  produced  by  the  multiplication  of  the  term  of  the  multi- 
plicand,  affected  with  the  highest  exponent  of  a  certain  letter,  by  the 
term  of  the  multiplier,  affected  with  the  highest  exponent  of  the 
same  letter.  2d.  The  term  produced  by  the  multiplication  of  the 
terms  affected  with  the  lowest  exponents  of  the  same  letter.  For 
those  two  partial  products  will  contain  this  letter,  affected  with  a 


MULTIPLICATION.  31 

higher  or  lower  exponent  than  either  of  the  other  partial  products, 
and  consequently  cannot  be  similar  to  any  of  them.  This  remark, 
the  truth  of  which  is  deduced  from  the  rule  of  the  exponents,  will 
be  very  useful  in  division. 

46.  To  finish  with  what  has  reference  to  algebraic  multiplication, 
we  will  make  known  a  few  results  of  frequent  use  m  algebra. 

1st,  Let  it  be  required  to  form  the  square  or  second  power  of  the 
binomial,     (a-\-b).     We  have, from  known  principles, 

(a+by=(a-{-b)  (a+b)  =  a''  +  2ab-j-Ir'. 
That  is,  the  square  of  the  sum  of  two  quantities  is  composed  of  the 
square  of  the  first,  plus  twice  the  product  of  the  first  by  the  second, 
plus  the  square  of  the  second. 

Thus,  lo  form  the  square  of  5a^-{-8a%  we  have,  from  what  has 
just  been  said, 

{5a'-\-8a'by=25a'-\-80a*b-i-64:aW. 

2d.   To  form  the  square  of  a  difference,  a—b,  w'c  have 
(a-by={a-b)  {a-b)=a'-2ab+^ 
That  is,  the  square  of  the  difference  between  two  quantities  is  com- 
posed of  the  square  of  the  first,  minus  twice  the  product  of  the  first 
by  the  second,  plus  the  square  of  the  second. 

Thus,  {laW—12aFy^4:9a'b'—168ab'+lUa'IP. 

3d.  Let  it  be  required  to  multiply  a-j-b  by  a—b. 

We  have  (a+5)x(a-5)=a2-^ 

Hence,  the  sum  of  two  quantities,  multiplied  by  their  difference, 
gives  the  difference  of  their  squares  for  a  product. 

Thus,         {Sa'+7alr)  (8a''-7aIr)=64:a^-4:9aW. 

We  can,  by  combining  these  different  results,  find  the  products 
of  certain  polynomials  more  promptly,  than  by  the  common  process. 
For  example,  multiply  5a^— 4aJ+33^,  by  ba^—Aab—dP.  If  we 
observe  that  the  first  of  these  two  quantities  is  the  sum  of  the  two 
quantities  ba^—4tab,  and  Sb%  and  that  the  second  is  the  difference  of 
the  same  quantities,  we  find  immediately  that  the  product  is 
{5a^-4aby-{Uy^25a'-^0ab  +  16aW-9bK 

47.  By  reflecting  upon  the  results  of  multiplication  that  we  have 
2 


32  ALGEBRA. 

just  obtained,  it  will  be  perceived  that  their  composition,  or  the 
manner  in  which  they  are  formed  from  the  multiplicand  and  multi- 
plier,  Is  entirely  independent  of  any  particular  values  that  may  be 
attributed  to  the  letters  a  and  h  which  enter  the  two  factors. 

The  manner  in  which  an  algebraic  product  is  formed  from  its  two 
factors,  is  called  the  law  of  this  product ;  and  this  law  remains  al- 
ways the  same,  whatever  values  may  be  attributed  to  the  letters 
which  enter  into  the  two  factors. 

48.  Lastly,  a  polynomial  being  given,  it  may  sometimes  be  de- 
composed into  factors  merely  by  inspection. 

Take  for  example  the  polynomial      aWc-\-bah^  -^ac. 

It  is  plain  that  «  is  a  factor  of  all  the  terms.  Hence,  we  may 
write  ahx+oaF +ac—a{1rc-{-fih^  +c). 

Take  the  polynomial  2ba^—Wa/h-\-\baW,  it  is  evident  that  5  and 
a^  are  factors  of  each  of  the  terms.  We  may,  therefore,  put  the 
polynomial  under  the  form      f)a\ba'—Qah-\-ZW). 

In  the  same  way  Q^a^¥ —Iba^W  is  transformed  into 

DIVISION. 

49.  Algebraic  division  has  the  same  object  as  arithmetical,  viz. 
having  given  a  product,and  one  of  its  factors,to  find  the  other  factor. 

We  will  first  consider  the  case  of  two  monomials. 

72a' 

The  division  of  72«'*  bv  8rt^  is  indicated  thus  :  

8a^ 

It  is  required  to  find  a  third  monomial,  which,  multiplied  by  the 
second,  will  produce  the  first.  Now,  by  the  rules  for  the  multipli- 
cation  of  monomials,  the  required  quantity  must  be  such  that  its  co- 
efficient  multiplied  by  8  should  give  72  for  a  product,  and  that  the 
exponent  of  the  letter  a  in  this  quantity,  added  to  3,  the  exponent  of 
the  letter  a  in  the  divisor,  should  give  5,  the  exponent  of  a  in  the  divi- 
dend. This  quantity  may,  therefore,  be  obtained  by  dividing  72  by 
8  and  subtracting  the  exponent  3  from  the  exponent  6, 


DIVISION.  33 

which  gives ~8a^~^^  * 

Tab 
for,  7aJx5a'5c=:35a^52c. 

50.   Hence  for  the  division  of  monomials  we  have  the  following 
RULE. 

I.  Divide  the  co-efficient  of  the  dividend  by  the  co-efficient  of  the 
divisor. 

II.  Write  in  the  quotient,  after  the  co-efficient,  all  the  letters  common 
to  the  dividend  and  divisor,  and  affect  each  with  an  exponent  equal  to 
the  excess  of  its  exponent  in  the  dividend  over  that  in  the  divisor. 

III.  Annex  to  these,  those  leUers  of  the  dividend,  with  their  re- 
spective exponents,  which  are  not  found  in  the  divisor. 

From  these  rules  we  find, 

ASa^'Prd  IbOa'b'cd'         „„    , 

-Aa-hcd  ;-:rr-TTrT7-=5a^4''crf. 


I2a¥c         '  ZQa'lfd? 

1.  Divide  IQ3?  by  8a;.  Ans.     2x. 

2.  Divide  I5axf  by  Say.  Ans.     bxf. 

3.  Divide  Ma¥x  by  12^'^  Ans.     labx. 
51.  It  follows  from  the  preceding  rule  that  the  division  of  mono- 

mials  will  be  impossible, 

1st.  When  the  co-efficients  are  not  divisible  by  each  other. 

2d.  When  the  exponents  of  the  same  letter  are  greater  in  the 
divisor  than  in  the  dividend. 

3d.  When  the  divisor  contains  one  or  more  letters  which  are  not 
found  in  the  dividend. 

When  either  of  these  three  cases,  occurs,  the  quotient  remains  un- 
der the  form  of  a  monomial  fraction,  that  is,  a  monomial  expression, 
necessarily  containing  the  algebraic  sign  of  division ;  but  which 
may  frequently  be  reduced. 

Take  for  example,  12a^lrcd  to  be  divided  by  Sa^bc^ 

Here  an  entire  monomial  cannot  be  obtained  for  a  quotient ;  that 


34  ALGEBRA. 

is  to  say,  a  monomial  which  does  not  contain  the  sign  of  division ; 

for  12  is  not  divisible  by  8,  and  moreover,  the  exponent  of  c  is  less 

in  the  dividend  than  in  the  divisor ;  therefore,  the  quotient  is  pre- 

,       ,       ,  12a'b^cd 

sented  under  the  form  ;     but  this  expression  can  be 

reduced,  by  observing  that  the  factors  4,  a^,  b  and  c  being  common 
to  the  two  terms  of  the  fraction,  may  be  suppressed,  and    we   have 

— - —     for  the  result. 
2c 

In  general,  to  reduce  a  monomial  fraction  it  is  necessary 
1st.    To  suppress  the  greatest  factor  common  to  tlie  tivo  co-efficienls, 
2d.  Subtract  the  less  of  the  two  exponents  of  the  same  letter,  from 
the  greater,  and  write  the  letter  affected  with  this  difference,  in  that 
term  of  the  fraction  corresponding  with  the  greatest  exponent. 

3d.   Write  those  letters  which  are  not  common,  with  their  respective 
exponents,  in  the  term  of  the  fraction  which  contains  the^n. 
From  this  new  rule,  we  find, 

^.SaWcd''       4.ad}  SlaPc'd      SWc 

and 


also. 


SGa^'cMe       Sbce  Ga'b  c'd?        6a'd' 

la%  1 


Wa^V        2ab 

In  the  last  example,  as  all  the  factors  of  the  dividend  are  found  in 
tlie  divisor,  the  numerator  is  reduced  to  rmity ;  for  it  amounts  to 
dividing  both  terms  of  the  fraction  by  the  numerator. 

52.  It  often  happens,  that  the  exponents  of  certain  letters,  are  the 
same  in  the  dividend  and  divisor. 

For  example,  divide  24a^Zr,  by  8a^/r ;  as  the  letter  b  is  affected 

with  the  same  exponent,  it  should  not  be  contained  in  the  quotient, 

2Aa''lr' 
and  we  have  — --Tr;-  =  3ff.     But  it  is  to  be  remarked,  that  this  re- 

suit,  Sa,  can  be  put  under  a  form  which  will  preserve  the  trace  of 
the  letter  b,  this  letter  having  disappeared  in  consequence  of  the 
reduction. 


DIVISION.  35 

For  if  we  apply,   conventionally,  the  rule   for   the   exponents, 

W  b- 

(Art.  50.),  to  the  expression  -^ — ,   it  becomes    ——b^-'=b'':  this 

new  symbol  b",  indicates  that  the  letter  enters  0  times,  as  a  iactor  in 
the  quotient  (Art.  13)  ;  or,  which  amounts  to  the  same  thing,  that  it 
does  not  enter  it ;  but  it  indicates  at  the  same  time,  that  it  was  in 
the  dividend  and  divisor,  and  that  it  has  disappeared  in  consequence 
of  the  operation.  This  symbol  has  the  advantage  of  preserving  the 
trace  of  a  quantity  which  constitutes  a  part  of  the  question,  that  it 
has  been  our  object  to  resolve,  without  changing  the  value  of  the 

result ;  for  since  b"  is  equivalent  to  — ,  which  is,  moreover,  equiva- 
lent to  1,  it  follows  that  SaJ^^Sax  l  =  3a.     In  like  manner, 

,  „    ^  —  Sa^yc"— 5Zr, 
3a^b  cr 

53.  As  it  is  important  to  have  clear  ideas  of  the  origin  and  significa- 
tion  of  the  symbols  employed  in  algebra,  we  will  show  that  in  gene- 
ral every  quantity  a  affected  with  the  exponent  0,  is  equivalent  to  1 ; 
that  is,  we  will  have    0"=!. 

For  this  expression  arises,  as  has  just  been  said,  from  the  fact  that 
a  is  affected  with  the  same  exponent  in  the  divisor  and  dividend. 

To  make  the  case  general,  let  m  denote  the  entire  number  which 

a"* 

IS  the  exponent  of  a.     We  shall   then  have,     —;^z=(f.     But  the 

quotient  of  any  quantity  divided  by  itself,  is  1.     Hence,     — =1  ; 

therefore,  we  also  have    «"=! 

We  observe  again,  that  the  symbol  a"  is  only  employed  conven- 
tionally, to  preserve  in  the  calculation  the  trace  of  a  letter  which 
entered  in  the  enunciation  of  a  question,  but  which  must  disappear  in- 
consequence of  a  division  ;  and  it  is  often  necessary  to  preserve  this 
trace. 


36  ALGEBRA. 

Division  of  Polynomials. 

54.   Let  it  be  required  to  divide  51a'&-+10a^  — iSa^Z*  — 15J'  +  4aJ^ 
by  4a^— 5(r4-36^     In  order  that  we  may  follow  the  steps  of  the 
operation  more  easily,  we  will  arrange  the  quantities  thus. 
Dividend.  Divisor. 

10a*-48a^6  +51a-&-  +  4aJ^-15iMl -5a=+4rt*+3^^ 

+  10a*-   Qa'b  -  QaW  -2  a'+Sab-blr' 

-  40a''b+  d7aW-\-^aI?'-15b*  Q^^otient. 

—  40a"J+  32a^^-+24«/;' 

2?>aIf-2Qah'-lbb' 
2r:>aW—20ab'  —  15b* 


The  object  of  this  operation  is,  as  we  have  already  said  (Art.  49), 
to  find  a  third  polynomial,  which,  multiplied  by  the  second,  shall 
produce  the  first. 

It  follows  from  the  definition  and  the  rule  for  the  multiplication 
of  polynomials  (Art.  43),  that  the  dividend  is  the  assemblage,  after 
addition  and  reduction,  of  the  partial  products  of  each  term  of  the 
divisor,  multiplied  by  each  term  of  the  quotient  sought.  Hence,  if 
we  could  discover  a  term  in  the  dividend  which  was  derived,  with- 
out reduction,  from  the  multiplication  of  one  of  the  terms  of  the 
divisor,  by  a  term  of  the  quotient,  then,  by  dividing  the  term  of  the 
dividend  by  that  of  the  divisor,  we  would  obtain  a  term  of  the  re- 
quired quotient. 

Now,  from  the  third  remark  of  Art.  45,  the  term  lOa",  affected 
with  the  highest  exponent  of  the  letter  a,  is  derived,  without  reduc- 
tion from  the  two  terms  of  the  divisor  and  quotient,  affected  with 
the  highest  exponent  of  the  same  letter.  Hence,  by  dividijig  the 
term  lOo*  by  the  term  —5a\  we  will  have  a  term  of  the  required 
quotient.  But  here  another  difficulty  presents  itself,  viz.  to  deter- 
mine the  sign  with  which  the  term  of  the  quotient  should  be  affl-cted. 
In  order  that  this  subject  may  not  impede  our  progress  hereafter, 
we  will  establish  a  rule  for  the'  signs  in  division. 


DIVISION.  37 

Since,  in  multiplication,  the  product  of  two  terms  having  the  same 
sign  is  affected  with  the  sign  +,  and  the  product  of  two  terms 
having  contrary  signs  is  affected  with  the  sign  — ,  we  may  con- 
clude, 

1st.  That  when  the  term  of  the  dividend  has  the  sign  +,  and 
that  of  the  divisor  the  sign  -f,  the  term  of  the  quotient  must  have 
the  sign  +. 

2d.  When  the  term  of  the  dividend  has  the  sign  +,  and  that  of 
the  divisor  the  sign  — ,  the  term  of  the  quotient  must  have  the 
sign  — ,  because  it  is  only  the  sign  — ,  which,  combined  with  the 
sign  — ,  can  produce  the  sign  +  of  the  dividend. 

3d.  When  the  term  of  the  dividend  has  the  sign  — ,  and  that  of  the 
divisor  the  sign  +,  the  quotient  must  have  the  sign  — . 

That  is,  when  the  two  terms  of  the  dividend  and  divisor  have  the 

same  sign,  the  quotient  will  be  affected  with  the  sign  +,  and  when 

they  are  affected  with  contrary  signs,  the  quotient  will  be  affected 

with  the  sign  —  ;  again,  for  the  sake  of  brevity,  we  say  that 

+  divided  by  +,  and  —  divided  by  — ,  give  +  ; 

—  divided  by  +»  and  +  divided  by  — ,  give  — . 

In  the  proposed  example,  10a*  and  —  5a^  being  affected  with 
contrary  signs,  their  quotient  will  have  the  sign  —  ;  moreover, 
I0a\  divided  by  5a%  gives  2a^ ;  hence,  —2a-  is  a  term  of  the  re- 
quired  quotient.  After  having  written  it  under  the  divisor,  multiply 
each  term  of  the  divisor  by  it,  and  subtract  the  product, 

from  the  dividend,  which  is  done  by  writing  it  below  the  dividend, 
conceiving  the  signs  to  be  changed,  and  performing  the  reduction. 
Thus,  the  result  of  the  first  partial  operation  is 

-40a^S+57a-6--f  4fl&'-  15Z»*. 

This  result  is  composed  of  the  partial  products  of  each  term  of 

the  divisor,  by  all  the  terms  of  the  quotient  which  remain  to  be  de- 

termined.     We  may  then  consider  it  as  a  new  dividend,  and  reason 

upon  it  as  upon  the  proposed  dividend.     We  will  therefore  take  in 


38  ALGEBRA. 

this  result,  the  term  —  40a'&,  affected  with  the  highest  exponent  of 
fl,  and  divide  it  by  the  tenn  —  5a^  of  the  divisor.  Now,  from  the 
preceding  principles,  —40a'*,  divided  by  —5a^  gives  -{-8ai  for  a 
new  term  of  the  quotient,  which  is  written  on  the  right  of  the  first. 
Multiplying  each  term  of  the  divisor  by  this  term,  and  writing  the 
products  underneath  the  second  dividend,  and  making  the  subtrac- 
tion, the  result  of  the  second  operation  is 

25d'Ir-20aP-15¥; 
then  dividing  25aW  by  —5a",  we  have  —5b^  for  the  third  term  of 
the  quotient.  Multiplying  the  divisor  by  this  term,  and  writing  the 
terms  of  the  product  under  the  third  dividend,  and  reducing,  we  ob- 
tain  0  for  the  result.  Hence,  —2a^  -i-&ab—5b%  or  Sab— 20^—56" 
is  the  required  quotient,  which  may  be  verified  by  multiplying  the 
divisor  by  it ;  the  product  should  be  equal  to  the  dividend. 

By  reflecting  upon  the  preceding  reasoning,  it  will  be  perceived, 
that,  in  each  partial  operation,  we  divide  that  term  of  the  dividend 
which  is  affected  with  the  highest  exponent  of  one  of  the  letters,  by 
that  term  of  the  divisor  affected  with  the  highest  exponent  of  the 
same  letter.  Now,  we  avoid  the  trouble  of  looking  out  the  term, 
by  taking  care,  in  the  first  place,  to  write  the  terms  of  the  dividend 
and  divisor  in  such  a  manner  that  the  exponents  of  the  same  letter  shall 
go  on  diminishing  from  left  to  right.  This  is  what  is  called  arrung. 
ing  the  dividend  and  divisor  with  reference  to  a  certain  letter.  By 
this  preparation,  the  first  term  on  the  left  of  the  dividend,  and  the 
first  on  the  left  of  the  divisor,  are  always  the  two  which  must  be 
divided  by  each  other  in  order  to  obtain  a  term  of  the  quotient ;  and 
it  is  the  same  in  all  the  following  operations  ;  because  the  partial 
quotients,  and  the  products  of  the  divisor  by  these  quotients  are 
always  arranged. 

55.  Hence,  for  the  division  polynomials  we  have  the  following 

RULE. 

I.  Arrange  the  dividend  and  divisor  with  reference  to  a  certain  letter, 
apd  then  divide  the  first  term  on  the  left  of  the  dividend  hy  the  first  term 


mvisioN.  39 

on  the  left  of  the  divisor,  the  result  is  the  first  term  of  the  quotient  ,• 
multiply  the  divisor  by  this  term,  and  subtract  the  'product  from  the 
dividend. 

II.  Then  divide  the  first  term  of  the  remainder  hy  the  first  term  of 
the  divisor,  which  gives  the  second  term  of  the  quotient ;  multiply  the 
divisor  by  this  second  term,  and  subtract  the  product  from  the  result  of 
the  first  operation.  Continue  the  same  process  until  you  obtain  Ofor 
a  resuU  ;  in  which  case  the  division  is  said  to  be  exact. 

When  the  first  term  of  the  arranged  dividend  is  not  exactly  divisi- 
ble by  that  of  the  arranged  divisor,  the  complete  division  is  impossi- 
ble, that  is  to  say,  there  is  not  a  polynomial  which,  multiplied  by 
the  divisor,  will  produce  the  dividend.  And  in  general,  we  will  find 
that  a  division  is  impossible,  when  the  first  term  of  one  o£  the  partial 
dividends  is  not  divisible  by  the  first  term  of  the  divisor. 

56.  Though  there  is  some  analogy  between  arithmetical  and 
algebraical  division,  with  respect  to  the  manner  in  which  the  opera- 
tions are  disposed  and  performed,  yet  there  is  this  essential  difference 
between  them,  that  in  arithmetical  division  the  figures  of  the  quo- 
tient  are  obtained  by  trial,  while  in  algebraical  division  the  quotient 
obtained  by  dividing  the  first  term  of  the  partial  dividend  by  the 
first  term  of  the  divisor  is  always  one  of  the  terms  of  the  quotient 
sought. 

Besides,  nothing  prevents  our  commencing  the  operation  at  the 
right  instead  of  the  left,  since  it  might  be  performed  upon  the  terms 
affected  with  the  lowest  exponent  of  the  letter,  with  reference  to 
which  the  arrangement  has  been  made.  In  arithmetical  division 
the  quotient  can  only  be  obtained  by  commencing  on  the  lefl. 

Lastly,  so  independent  are  the  partial  operations  required  by  the 
process,  that  after  having  subtracted  the  product  of  the  divisor  by 
the  first  term  found  in  the  quotient,  we  could  obtain  another  term  of 
the  quotient  by  dividing  by  each  other  the  two  terms  of  the  new  divi- 
dend and  divisor,  affected  with  the  highest  exponent  of  a  different 
letter  from  the  one  first  considered.  If  th^same  letter  is  preserved, 
it  is  because  there  is  no  reason  for  changing  it,  and  because  the  two 


40  ALGEBRA. 

polynomials  are  already  arranged  with  reference  to  it;  the  first 
terms  on  the  left  of  the  dividend  and  divisor  being  sufficient  to  obtain 
a  term  of  the  quotient ;  whereas,  if  the  letter  is  changed,  it  would 
be  necessary  to  seek  again  for  the  highest  exponent  of  this  letter. 

SECOND  EXAMPLE. 

Divide     .     .     .     21a;y+25a;y+68ay-40/— 56x'— 18a:Vby 
5y''  —  8x''  —  6xy. 

—  ^0y'+68xy'  +  25xy+21xy-18x*y-56x'\\5y''-6xy-8x"- 

—  40y^- + 48^^ + 64A-y  —8y'+Axy?-3x-y+lx^ 
1st.  rem.  20  a,y— 39xy +21a;y 

2  0  xy' — 24x-'^'  —  32  xY 
2d.  rem.  —  15a;y   +53xy  — ISa;^ 

—  Wxy  4-1 8x^3;=' + 24:X*y 

35xy-  ^2x*y-56x' 
35a,y—  42x'^y-56x' 


Final  remainder  0. 

57.  Remark. — In  performing  the  division,  it  is  not  necessary  to 
bring  down  all  the  terms  of  the  dividend  to  form  the  first  remainder, 
but  they  may  be  brought  down  in  succession,  as  in  the  example. 

As  it  is  important  that  beginners  should  render  themselves  familiar 
with  the  algebraic  operations,  and  acquire  the  habit  of  calculating 
promptly,  we  will  treat  of  this  last  example  in  a  different  manner, 
at  the  same  time  indicating  the  simplifications  which  should*  be" 
introduced. 

As  in  arithmetic,  they  consist  in  subtracting  each  partial  product 
from  the  dividend  as  soon  as  this  product  is  formed. 
—  40j/' + 68xy' + 25a''2/' +21  ^Y  —  1  Sx*y — 56a;'  ||  5?/'  —  dxy — 8x^ 
1st.  rem.  20a;/  — 39xy+21  a^y  —  8j/=+4a,y— 3ar'y+7x' 

2d.  rem.  —iSxy-f-  53  xY  —  18x*y 

3d.  rem.  4  ^^  a;y  — 42a;'7/— 56x' 

Final  rem.  0. 


DIVISION.  41 

First,  by  dividing  —40^^  by  5y^,  we  obtain  —8/  for  the  quotient. 
Multiplying  5?/^  by  —  8^,  we  have  —40?/*,  or  by  changing  the  sign, 
^-402/^  which  destroys  the  first  term  of  the  dividend. 

In  like  manner,  —6xyX  —^y^  gives  +4:8xy*  and  for  the  subtrac- 
tion —  48a^*,  which  reduced  with  -\-68xy\  gives  20xy*  for  a  remain- 
der. Again,  —Svc'^X— 8^'  gives +,  and  changing  sign,  —  64a;y, 
which  reduced  with  25x'y^,  gives  —  39a;y.  Hence  the  result  of  the 
first  operation  is  20xy*—S9xY  followed  by  those  terms  of  the  divi- 
dend  which  have  not  been  reduced  with  the  partial  products  already 
obtained.  For  the  second  part  of  the  operation,  it  is  only  necessary 
to  bring  down  the  next  term  of  the  dividend,  separating  this  new 
dividend  from  the  primitive  by  a  line,  and  operate  upon  this  new 
dividend  in  the  same  manner  as  we  operated  upon  the  primitive,  and 
so  on. 

THIRD  EXAMPLE. 

Divide     95a-73a'+56a^-25-59a^  by  -2a^+5-Ua-\-la' 
56a*  — 59a'  — 73a'  +  95fl— 25||7a'  — 3a'  — lla+5 
1st.  rem.       ^-3^5a^+lba'-\-b5a—25  8  a  —  5 

2d.  rem.  0. 

EXAMPLES. 

1.  Divide  ISa;"  by. 9a;.  Ans.     2x. 

2.  Divide  lOx^  by   —bx'y.  Ans.     —2y. 

3.  Divide  —  9axy  by  Qx'^y.  Ans.     —ay. 

4.  Divide  —  8x^  by  —2x.  Ans.     -|-4r. 

5.  Divide  10a5+15ac  by  5a.  Ans.     23-j-3c. 

6.  Divide  30aa;— 54a;  by  6a;.  Ans.     5a— 9. 

7.  Divide  \Ox^y—\by''—by  by  5t/.  Ans.     2x'—dy—l. 

8.  Divide  13a+3aa?— 17a;''  by  21a. 

9.  Divide  Sa"  — 15-t-6a+3J  by  3a. 

10.  Divide  a''+2aa;-l-a;''  by  a-\-x.  Ans.     a-\-x. 

11.  Divide  a^  —  Za^y-\-^ay'^—y'^  by  a—y. 

Ans.     a'— 2ay+w', 

4* 


42  ALGEBRA. 

12.  Divide  1  by  1— x.  Ans.     l+x+x^+a;',  &c. 

13.  Divide  6a;'  — 96  by  3x— 6.  Ans.  2a;'+4x''+8x+16. 

14.  Divide  a'  — 5a*x4-10aV  — 10aV+5aa;*— «' by  a'— 2aa;+x'. 

Ans.     a^—Za^x+^ax^—x^. 

15.  Divide  48a;=— 76aa;''  — 64a'a;+105a^  by  2a;— 3a. 

16.  Divide/-3i/V432/V— a;^  by  y'—^''x+Zyx''—x\ 

58.  It  may  happen  that  one,  or  both,  of  the  proposed  polynomials 
contains  in  two  or  more  temis  the  same  power  of  the  letter  with  re- 
ference to  which  the  arrangement  is  to  be  made. 

In  this  case,  how  should  the  arrangement  be  made,  and  the  divi- 
sion be  effected  1 

Divide     lla^i-19aJc  +  10a'-15a^c+3aJ^  +  15Jc^-5&^c 
by        5a'-f3a&— 5Jc. 

In  the  first  place,  the  two  terms  lla^J— ISa^'c,  can  be  placed  un- 
der the  form  (11&— 15c)  a",  or     Hi  I  a",  by  writing  the  power  a' 

-15c  1 
once,  and  placing  to  the  left  of  it,  and  in  the  same  vertical  column, 
the  quantities  by  which  this  power  is  multiplied ;  this  polynomial 
multiplier  is  then  called  the  co-efficient  of  a^. 

The  second  manner  of  connecting  the  terms  involving  the  same 
power,  is  preferable  to  the  first,  for  two  reasons.  1st.  Because 
where  there  are  many  terms  in  the  dividend  and  divisor,  it  would  be 
difficult  to  write  all  on  the  same  horizontal  line.  2d.  As  the  co-ef- 
ficient  of  each  power  ought  to  be  arranged  with  reference  to  a 
second  letter,  we  are  obliged,  if  the  first  term  is  subtractive,  to  sub- 
ject the  term  to  a  modification,  which  might  lead  to  error,  in  employ- 
ing the  first  manner.  Take,  for  example,  —\blPa^ -\-lhca^ —  Qc^a' 
the  modification  consists  in  putting  this  expression  under  the  form 

—  {lbb''  —  'Jbc+Sc')a'     ....     (Art.  38). 
whereas,  by  the  second,  it  is  written  thus  :     —Ibh"^     a",  and  by  this 

+  lie 

—  ^e 

manner  we  have  the  advantage  of  preserving  to  each  term  the  sign 
with  which  it  was  at  first  affected. 


DIVISION.  43 

In  like  manner,  —l9abc-{-Sah^  is  written  :  .     .      -\-  Sb^    \  a 

-19bc  I 
This  being  understood  the  operation  may  be  performed  in  the 
following  manner. 

-15c|    -19^1  '2a+b-3c 

a—5¥c+l5b(? 


1st.  Rem.  5b 

—  15c 


-9bc 


2d.  Rem.  0. 

First  divide  10  a^  by  5a-,  the  quotient  is  2a.  Subtracting  the 
product  of  the  divisor  by  '2a,  we  obtain  the  first  remainder.  Divi- 
ding the  part  involving  a'  in  this  remainder  by  5a^,  the  quotient  is 
b — 3c.  Multiplying  successively  each  term  of  the  divisor  by  b—3c, 
and  subtracting  the  product,  we  have  0  for  the  result.  Hence, 
2a 4-^  — 3c  is  the  required  quotient. 

59.  Among  the  different  examples  of  algebraic  division,  there  is 
one  remarkable  for  its  applications.  It  is  so  often  met  with  in  the 
resolution  of  questions,  that  algebraists  have  mg.de  a  kind  oHheorem 
of  it. 

We  have  seen  (Arj^  46),  that 

{a-\-by\a—b)  ^a^  —  J^:         hence, 

a^-J2 

--a-\-b. 


a  —b 
If  we  divide     .     .     a^—b^     by     a—b     we  have 


a^-b-" 

a — b 
a*—¥ 


-.a^-\-ab-\-b'^  :  also 

:a"+a'3+a5^+i= 


a  —b 
by  performing  the  division. 

These  are  results  that  may  be  obtained  by  the  ordinary  pro 
cess  of  division.  Analogy  would  lead  to  the  conclusion  that  what- 
ever  may  be  the  exponents  of  the  letters  a  and  b,  the  division  could 
be  performed  exactly ;  but  analogy  does  not  always  lead  to  cer- 


44 


ALGEBRA. 


tainty.     To  be  certain  on  this  point,  denote  the  exponent  by  m  ;  and 
proceed  to  divide  a"'—b'"  by  a  —  b. 

a"'—b'^       \\a—h 


1st.  Rem.     .     .     .     a"^^b—b"'\a'"--^  + 
or         ....     3(a"*-'-J'»-'). 

Dividing  a"  by  a  the  quotient  is  a"'-\  by  the  rule  for  the  exponents. 
The  product  of  a— 3  by  a™-'  being  subtracted  from  the  dividend,  the 
first  remainder  is  a'^'i— J",  which  can  be  put  under  the  form 
b  (a"'-^—b'"-^).  Now,  if  «"-'  —  §'»-'  is  divisible  by  a—b,  then  will 
oT'—b'"  also  be  divisible  by  a— ^^ ;  that  is,  if  the  difference  of  the 
similar  powers  of  two  quantities  of  a  certain  degree,  is  exactly  divisi- 
ble  by  the  difference  of  these  quantities,  the  difference  of  the  powers 
of  a  degree  greater  by  unity,  is  aJso  divisible  by  it. 

But  it  has  already  been  shown  that  a*— i*  is  divisible  hy  a—b  : 
hence,  a^—b^  is  also  divisible  by  a—b.  Now,  if  a^—¥  is  divisible 
hy  a—b,  it  must  follow  that  a^  —  b''  is  also  divisible  hy  a—b.  In  the 
same  way  ;t  may  be  shown  that  the  division  is  possible  when  the 
exponent  is  7,  8,  9,  &c. 

Hence,  generally,  a*"— J'"  is  divisible  hy  a—b. 

This  proposition  may  be  verified  by  actually  performing  the 
division,  and  then  multiplying  the  quotient  by  the  divisor.     Thus, 


a  -b    - 
But     .     . 

ultiplied  by 

.     a'^-'+a^-^b+a'^-'b-'     . 
.     a  -    b 

_    a^-^h-a'"-^¥     . 

.     .        —ab^-'  —  b'". 

equal  to     . 

.     .                    a"—  *'". 

It  will  be  perceived  that  the  partial  products  a"  and  —  &"•  are  the 
only  ones  that  do  not  destroy  each  other  in  the  reduction. 

For  example,  multiplying  a""-^b  by  a,  the  product  is  a'"-'3  ;  but 
by  multiplying  a"*^'  by  —b,  the  product  is  —a"'-%  and  this  term 
destroys  the  preceding.     The  other  terms  cancel  in  the  same  way. 


DIVISION.  45 

The  beginner  should  reflect  upon  the  first  method  of  demonstrating 
the  proposition,  as  it  is  frequently  employed  in  algebra. 

60.  We  have  given  (Art.  51.  and  55.),  the  principal  circum- 
stances by  which  it  may  be  discovered  that  the  division  of  monomial 
or  polynomial  quantities  is  not  exact;  that  is,  the  case  in  which 
there  does  not  exist  a  third  entire  algebraic  quantity,  which,  multi- 
plied by  the  second,  will  produce  the  first. 

We  will  add,  as  to  polynomials,  that  it  may  often  be  discovered 
by  mere  inspection  that  they  cannot  be  divided  by  each  other. 
When  these  polynomials  contain  two  or  more  letters,  before  arrang- 
ing  them  with  reference  to  a  particular  letter,  observe  the  two 
terms  of  the  dividend  and  divisor,  which  are  affected  with  the 
highest  exponent  of  each  of  the  letters.  If  for  either  of  these  let- 
ters,  one  of  the  terms  with  the  highest  exponent  is  not  divisible  by  the 
other,  we  may  conclude  that  the  total  division  is  impossible.  This 
remark  applies  to  each  of  the  operations  required  by  the  process 
for  finding  the  quotient. 

Take,  for  example,  12a^—da-h-\-lab''—Ub\  to  be  divided  by 
Aa'  —  8ab-{-Sb\ 

By  considering  only  the  letter  a,  the  division  would  appear  pos- 
sible ;  but  regarding  the  letter  b,  the  division  is  impossible,  since 
—  llb^  is  not  divisible  by  35\ 

One  polynomial  A,  carmot  be  divided  by  another  B  containing  a 
letter  which  is  not  found  in  the  dividend ;  for  it  is  impossible  that  a 
third  quantity, multiplied  by  B  which  depends  upon  a  certain  letter, 
should  give  a  product  independent  of  this  letter. 

A  monomial  is  never  divisible  by  a  polynomial,  because  every 
polynomial  multiplied  by  another,  gives  a  product  containing  at  least 
two  terms  which  are  not  susceptible  of  reduction. 

61.  Remark If  the  letter  with  reference  to  which  the  dividend 

is  arranged,  is  not  found  in  the  divisor,  the  divisor  is  said  to  be  inde- 
pendent of  that  letter ;  and  in  that  case  the  exact  division  is  impos- 
sible, unless  the  divisor  mill  divide  separately  the  co-efficient  of  each 
term  of  the  dividend. 


46  ALGEBRA. 

For  example,  if  the  dividend  were  Sba'  +  9ba^+12h,  arranged 
with  reference  to  the  letter  a,  and  the  divisor  3b,  the  divisor  would 
be  independent  of  the  letter  a ;  and  it  is  evident  that  the  exact  divi- 
sion could  not  be  performed  unless  the  co-efRcient  of  each  term  of 
the  dividend  were  divisible  by  3Z>.  The  exponents  of  the  leading 
letter  in  the  quotient  would  be  the  same  as  in  the  dividend. 

OF  ALGEBRAIC  FRACTIONS. 

62.  Algebraic  fractions  should  be  considered  in  the  same  point  of 
view  as  arithmetical  fractions,  such  as  f ,  |i,  that  is,  we  must  con- 
ceive that  the  unit  has  been  divided  into  as  many  equal  parts  as 
there  are  units  in  the  denominator,  and  that  one  of  these  parts  is 
taken  as  many  times  as  there  are  units  in  the  numerator.  Hence, 
addition,  subtraction,  multiplication,  and  division,  are  performed  ac- 
cording  to  the  rules  established  for  arithmetical  fractions. 

It  will  not,  therefore,  be  necessary  to  demonstrate  those  rules, 
and  in  their  application  we  must  follow  the  procedures  indicated  for 
the  calculus  of  entire  algebraic  quantities. 

63.  Every  quantity  which  is  not  expressed  under  a  fractional 
form  is  called  an  entire  algebraic  quantity. 

64.  An  algebraic  expression,  composed  partly  of  an  entire  quan- 
tity and  partly  of  a  fraction,  is  called  a  mixed  quantity. 

65.  When  a  division  of  monomial  or  poljniomial  quantities  cannot 
be  performed  exactly,  it  is  indicated  by  means  of  the  known  sign, 
and  in  this  case,  the  quotient  is  presented  under  the  form  of  a  frac- 
tion, which  we  have  already  learned  how  to  simplify  (Art.  51). 
With  respect  to  polynomial  fractions,  the  following  are  cases  which 
are  easily  reduced. 

a^'  —  b^ 
Take,  for  example,  the  expression     -; — — ^ — r— 
'  a-  —  2ab-{-b' 

This    fraction   can    take   the   form  ,       ,~        (Art.  46). 

(a— by  ' 


OF  FRACTIONS.  47 

Suppressing  the  factor  a  —  h,  which  is  common  to  the  two  terms, 

we  obtain    ....        •   , 
a—b 

5a^  —  10a^b+5ab^ 

Again,  take  the  expression     r-^ — 3-77 

oa  — Oft  0 

This  expression  can  be  decomposed  thus  :     — -j- rr 

ba(a  —  b)- 
or — — . 

8a\a-b) 

Suppressing  the  common  factor,  a(a—b,)  the  result  is     .     . 
5(a-b) 
8a     * 
The  particular  cases  examined  above,  are  those  in  which  the  two 
terms  of  the  fraction  can  be  decomposed  into  the  product  of  the  sum 
by  the  difference  of  two  quantities,  and  into  the  square  of  the  sum  or 
difference  of  two  quantities.     Practice  teaches  the  manner  of  per- 
forming these  decompositions,  when  they  are  possible. 

But  the  two  terms  of  the  fraction  may  be  more  complicated  poly- 
nomials, and  then,  their  decomposition  into  factors  not  being  so  easy, 
we  have  recourse  to  the  process  for  finding  the  greatest  common 
divisor. 

CASE  I. 

Of  the  Greatest  Common  Divisor. 

66.  The  greatest  common  divisor  of  two  polynomials,  is  the  great- 
est polynomial,  with  reference  to  the  exponents  and  co-efficients,  that 
will  exactly  divide  the  proposed  polynomials.  , 

If  two  polynomials  be  divided  by  their  greatest  common  divisor, 
the  quotients  will  he  prime  with  resjject  to  each  other ;  that  is,  they 
will  no  longer  contain  a  common  factor. 

For,  let  A  and  B  be  the  given  polynomials,  D  their  greatest  com- 
mon divisor,  A'  and  B'  the  quotients  afler  division*.     Then 

*  Note. — When  the  same  letter  is  used  to  designate  different  quantities,  as 
above,  the  quantities  having  a  certain  connexion  with  each  other,  we  read  A', 
B',  A  prime,  B  prime,  and  if  we  have  A",  B",  we  say,  A  second,  B  second,  &c. 


48  ALGEBRA. 

A  B 

-^=A'         and         -^=B' 

Or     .     .     .     A=A'xD  and  B=B'xD 
now  if  A'  and  B'  had  a   common  factor   d,  it   would   follow  that 
dxD   would  be  a  divisor,  common  to  the  two  polynomials,  and 
greater  than  D,  either  with  respect  to  the  exponents  or  the  co-effi- 
cients, which  would  be  contrary  to  the  definition. 

Again,  since  D  exactly  divides  A  and  B,  every  factor  of  D  will 
have  a  corresponding  factor  in  both  A  and  B.     Hence, 

1st.  The  greater  common  divisor  of  two  polynomials  contains  as 
factors,  all  the  particular  divisors  common  to  the  two  polynomials,  and 
does  not  contain  any  other  factors. 

67.  We  will  now  show  that  the  greatest  common  divisor  of  two 
polynomials  will  divide  their  remainder  after  division. 

Let  A  and  B  be  two  polynomials,  D  their  greatest  common  divisor, 
and  suppose  A  to  contain  the  highest  exponent  of  the  letter  with  re- 
ference to  which  they  are  arranged.     Then, 

A  B 

Yr=A'  and  — =B'         or, 

A=A'xD     and     B=B'xD. 
Let  us  now  represent  the  entire  part  of  the  quotient  by  Q  and  the 
remainder  by  R,  and  we  shall  have 

A      A'xD  R 

A'xD=B'xDxQ+R 
R 
hence,  A'=B'xQ+^. 

But  A'  is  an  entire  quantity,  hence  the  quantity  to  which  it  is 

R 

equal  is  also  entire  :  and  since  B'Q  is  entire,  it  follows,  that  —     is 

entire  ;  that  is,D  will  exactly  divide  R, 

We  will  now  show  that  if  D  will  exactly  divide  B  and  R  that  it 
will  also  divide  A.     For,  having  divided    A  by  B   we  have 


OF  FRACTIONS. 


49 


A=BxQ+R>  and  by  dividing  by  D,  we  obtain 
A    B      ^     R 

But  since  we  suppose  B  and  R  to  be  divisible  by  D,  and  know  Q 
to  be  an  entire  quantity,  the  second  part  of  the  equality  is  entire  ; 
hence  the  first  part,  to  which  it  is  equal,  is  also  entire  ;  that  is,  A  is 
exactly  divisible  by  D.     Hence, 

2dly .  The  greatest  common  divisor  of  tivo  polynomials  is  the  same 
as  that  which  exists  between  the  least  polynomial  and  their  remainder 
after  division. 

These  principles  being  established,  let  us  suppose  that  it  is  re- 
quired  to  find  the  greatest  common  divisor  between  the  two  poly- 
nomials 

a^  —  a'h  +  Mh'  —  U^  and  a'  — 5rt6+4J^ 


First  Operation. 

a'-a'b  ^Sai'-Sb'\\a'-bab  +  W' 

^a'b-ab^-2b'   \a+  U 

1st.  Rem. 

.     .              19ab'-19b' 

or     . 

.     .              19b%a-b) 

Second  Operation. 

a'-dab+^b'  \\a-b 

—  4:ab-\-U-  1  rt— 4i 

0. 

Hence,  a— Z»  is  the  greatest  common  divisor. 

We  begin  by  dividing  the  polynomial  of  the  highest  degree  by 
that  of  the  lowest  degree  ;  the  quotient  is,  as  we  see  in  the  above 
table,  a-\-Ab  and  the  remainder  is  19ab'^—19b^. 

By  the  second  principle,  the  required  common  divisor  is  the  same 
as  that  which  exists  between  this  remainder  and  the  polynomial 
divisor. 

But  19ab'^—l9¥  can  be  put  under  the  form  I9b-(a—b).  Now 
5 


50  ALGEBRA. 

the  factor,  19i^  will  divide  this  remainder  without  dividing 

a'  —  bah+4.h\ 

hence,  by  the  first  principle,  this  factor  cannot  enter  into  the  greatest 

common  divisor  ;  we  may  therefore  suppress  it,  and  the  question 

is  reduced  to  finding  the  greatest  common  divisor  between' 

a^  —  bah+lP  and  a—h. 

Dividing  the  first  of  these  two  polynomials  by  the  second,  there  is 
an  exact  quotient,  a  — 4Z>;  hence  a—b  is  their  greatest  common 
divisor,  and  is  consequently  the  greatest  common  divisor  of  the  two 
proposed  pohmomials. 

Again,  take  the  same  example,  and  arrange  the  polynomials  with 
reference  to  h. 

—  SJ'-I-SflJ'— a^^+a\  and  W-bab-\-a\ 

First  Ojoeration. 

lJ/-rl2ah'-4a'b+4:a'  ||  4  b''~5ab-\-a' 


1st.  Rem. 

—    dah'—a''b   +4a' 
-I2ah"--4:a'b+l6a' 

-Sb, 

-3a 

2d.  Rem.     . 
or 

-19a^^  +  19a^ 
19a\-b-{-a). 

Second  Operation. 

4P-^5ab+o'\\    -I 

+n 

—ab  +a'  I  —  4tb-{-a 
0. 
Hence,  —b-^a,  or  n  —  b,  is  the  greatest  common  divisor. 
Here  we  meet  with   a  difficulty  in  dividing  the  two  polymonials, 
because  the  first  term  of  the  dividend  is  not  exactly  divisible  by  the 
first  term  of  the  divisor.     But  if  we  observe  that  the  co-efficient  4  of 
this  last,  is  not  a  factor  of  all  the  terms  of  the  polynomial 

4^^  — 5rtZ>-|-a% 
and  that  therefore,  by  the  first  principle,  4  cannot  form  a  part  of  the 
greatest  common  divisor,  we  can,  without  afiecting  this  common 


OF  FRACTIONS.  51 

divisor,  introduce  this  flictor  into  the  dividend.     This  gives 

-123'  +  12aZ<=— 4a"5+4a% 
and  then  the  division  of  the  first  two  terms  is  possible. 

Effecting  this  division,  the  quotient  is  —  Sb,  and  the  remainder  is 
—  3ab^—a'b-i-4:a\ 

As  the  exponent  of  5  in  this  remainder  is  still  equal  to  that  of  the 
divisor,  the  division  may  be  continued,  by  multiplying  this  remainder 
by  4,  in  order  to  render  the  division  of  the  first  term  possible. 

This  done,  the  remainder  becomes  —  12a^^  — 4a^^+16a^  which 
divided  by  4.b^  —  dal)-\-a',  gives  the  quotient  —3a,  which  should 
be  separated  from  the  first  by  a  comma,  having  no  connexion  with  it; 
and  the  remainder  is       .     —  19a*J+19a^. 

Placing  this  last  remainder  under  the  form  19a-(  — 3+a),  and  sup- 
pressing  the  factor  19rt^,  as  forming  no  part  of  the  common  divisor, 
the  question  is  reduced  to  finding  the  greatest  common  divisor 
between  4:P  —  5ab-]-a',  and  •-b-\-a. 

Dividing  the  first  of  these  polynomials  by  the  second,  we  obtain  an 
exact  quotient,  — 45+a;  hence  —b+a,  or  a— b,  is  the  greatest 
common  divisor  requii'ed. 

68.  In  the  above  example,  as  in  all  those  in  which  the  exponent 
of  the  prmcipal  letter  is  greater  by  unity  in  the  dividend  than  in  the 
divisor,  we  can  abridge  the  operation  by  multiplying  every  term  of 
the  dividend  by  the  square  of  the  co-efiicient  of  the  first  term  of  the 
divisor.  We  may  easily  conceive  that,  by  this  means,  the  first  par- 
tial  quotient  obtained  will  contain  the  first  power  of  this  co-efficient. 
Multiplying  the  divisor  by  the  quotient,  and  making  the  reductions 
with  the  dividend  thus  prepared,  the  result  will  still  contain  the  co- 
efficient  as  a  factor,  and  the  division  can  be  continued  until  a  re- 
mainder is  obtained  of  a  lower  degree  than  the  divisor,  with  refe- 
rence to  the  principal  letter. 

Take  the  same  example  as  before,  viz.  —SP  +  Sab'^—a^b-i-a^ 
and  4Z>''  — 5rti+rt^ ;  and  multiply  the  dividend  by  the  square  of 
4=16  :     and  we  have 


52  ALGEBRA. 

First  Operation. 

-m¥  +  ^Qab-  -IQa'b   +    16a=||43=-5aZ>+a^ 
-12a//   —   Ad'b   4-    16a°|  — 126-3a 
1st.  Rem.     .     .  —  IQa'^Z*    +    lOa^ 

IQa'-'  (-^'+«) 

Second  Operation. 

Alf  —  bah+a^  \\  —h+a 
—  ab+a^  I  —^b+a 


or 


2d-  Rem.      ...  -  0. 

Remark  1.  When  the  exponent  of  the  principal  letter  in  the  di- 
vidend exceeds  that  of  the  same  letter  in  the  divisor  by  two,  three, 
&c.  units,  multiply  the  dividend  by  the  third,  fourth,  &c.  power  of 
the  co-efficient  of  the  first  term  of  the  divisor.  It  is  easy  to  see  the 
reason  of  this. 

2.  It  might  be  asked  if  the  suppression  of  the  factors,  common  to 
all  the  terms  of  one  of  the  remainders,  is  absolutely  necessary,  or 
whether  the  object  is  merely  to  render  the  operations  more  simple. 
Now,  it  will  easily  be  perceived  that  the  suppression  of  these  factors 
is  necessary  ;  for,  if  the  factor  19a^  was  not  suppressed  in  the  pre- 
ceding  example,  it  would  be  necessary  to  multiply  the  whole  divi- 
dend  by  this  factor,  in  order  to  render  the  first  term  of  the  dividend 
divisible  by  the  first  term  of  the  divisor ;  but  then,  a  factor  would 
be  introduced  into  the  dividend  which  was  also  contained  in  the  divi- 
sor ;  and  consequently  the  required  greatest  common  divisor  would 
be  combined  with  the  factor  19a^  which  should  not  form  a  part  of  it. 

69.  For  another  example,  it  is  proposed  to  find  the  greatest  com. 
mon  divisbr  between  the  two  polynomials, 

a'+^a'b+^a'b'-Qab'+^b'  and  Aa''b+2ab'-^b\ 
or  simply,  2a''+ab—b\  since  the  factor  2b  can  be  suppressed,  being 
a  factor  of  the  second  polynomial  and  not  of  the  first. 


OF  FRACTIONS. 


53 


First  Operation. 

8a* + 24a'J + 32a^5' - 48a&'  + 1 65*  1 1  '■Za^+ah-lf 
+20a='6+36a-'^''-48a5'  +  16F|  4a'  +  10ai+13A=' 
+  26a'Z»'  — 38a5'  +  16^'* 


1st.  Rem.      .     .     .  -b\aV  +  -2W 

or,  .     .     .  —¥{bla-2U). 

Second  Operation. 

Multiply  by  2601,  the  square  of  51. 

5202a^+2601aZ>-2601^>'||   51a-   295 
5202a^-2958a5  I  102a  +  109& 


1st.  Rem.     .  +5559a6- 26015^ 

5559ai-316lZ'' 


2d.  Rem.      ...  +  5606^ 

The  exponent  of  the  letter  a  in  the  dividend,  exceeding  that  of 
the  same  letter  in  the  divisor  by  two  units,  we  multiply  the  whole 
dividend  by  the  cube  of  2,  or  8.  This  done,  we  perform  three  con- 
secutive  divisions,  and  obtain  for  the   first   principal   remainder, 

-51a5'+29Z>*. 
Suppressing  b^  in  this  remainder,  it  becomes  — 51a+295  for  a  new 
divisor,  or,  changing  the  signs,  which  is  permitted,  51a— 295:  the 
new  dividend  is         ^a^-^-ah—h". 

Multiplying  this  dividend  by  the  square  of  51,  or  2601,  then  effect- 
ing the  division,  we  obtain  for  the  second  principal  remainder,  +5605^ 
which  proves  that  the  two  proposed  polynomials  are  prime  with  re- 
spect  to  each  other,  that  is,  they  have  not  a  common  factor.  In  fact 
it  results  from  the  second  principle  (Art.  67),  that  the  greatest  com- 
mon divisor  must  be  a  factor  of  the  remainder  of  each  operation ; 
therefore  it  should  divide  the  remainder  5605== ;  but  this  remainder 
is  independent  of  the  principal  letter  a ;  hence,  if  the  two  polyno- 
mials  have  a  common  divisor,  it  must  be  independent  of  a,  and  will 
consequently  be  found  as  a  factor  in  the  co-efficients  of  the  different 
powers  of  this  letter,  in  each  of  the  proposed  polynomials  ;  but  it  is 
5* 


54  ALGEBRA. 

evident  that  the  co-efficients  of  these  polynomials  have  not  a  com- 
mon factor. 

70.  These  examples  are  sufficient  to  point  out  the  course  the  be- 
ginner  is  to  pursue,  in  finding  the  greatest  common  divisor  of  two 
polynomials,  which  may  be  expressed  by  the  following  general 

RULE. 

I.  Take  the  first  polynomial  and  suppress  all  the  mo7iotniaI  factors 
common  to  each  of  its  terms.  Do  the  same  with  the  second  polynomial, 
and  if  the  factors  so  suppressed  have  a  common  divisor,  set  it  aside  as 

forming  a  part  of  the  com7no7i  divisor  sought. 

II.  Having  done  this,  prepare  the  dividend  in  a  such  a  manner 
that  its  first  term  shall  be  divisible  by  the  first  term  of  the  divisor ;  then 
perform  the  division,  which  gives  a  remainder  of  a  degree  less  than 
that  of  the  divisor,  in  which  suppress  all  the  factors  that  are  common 
to  the  co-efiicients  of  the  different  jfowers  of  the  j^rincipal  letter.  Then 
fake  this  remainder  as  a  divisor,  and  the  second  polyno^nial  as  a  divi- 
dend,  and  cojitinue  the  operation  with  these  polynomials,  in  the  same 
manner  as  with  the  p)receding. 

III.  Continue  this  series  of  operations  until  a  remainder  is  obtained 
which  will  exactly  divide  the  preceding  rcTnainder  ;  this  last  retnainder 
will  be  the  greatest  common  divisor ;  but  if  a  remainder  is  obtained 
which  is  independent  of  the  principal  letter,  and  which  will  not  divide 
the  co-efficients  of  each  of  the  proposed  polynomials,  it  shoics  that  the 
jiroposed  polynomials  are  prime  with  respect  to  each  other,  or  that 
they  have  not  a  common  factor. 

EXAMPLES, 

1.  Find  the  greatest  common  divisor  between  the  two  poly- 
nomials. 

ah-\-2a-  —  '&¥  —4bc—  ac  —  c\ 
and     .     .       9ac+2a'— 5aJ  +  4c=+8k  — 12^' 


OF  FRACTIONS. 


55 


First  Operation. 


^h 

a-U^ 

2a^-5b 

a- 125" 

—  c 

—Uc 

+  9c 

+  8bc 

—   c^ 

+   4c^ 

1st.  Remainder 


65 
10c 


a +95" 
-125c 
-  5  c* 
(35— 5c)  (2a+35+c). 

Second  Operation. 


2a^-55 
+9c 


a- 125" 

+85c 
+4c" 


-85 
+8c 


t-125= 
+  85c 
+  4c" 


2a+35+c 


s— 45 
+4c 


0. 

Hence,  2fl+ 35 +c  is  the  greatest  common  divisor. 
After  arranging  the  two  polynomials,  the  division  may  be  perform- 
ed  without  any  preparation,  and  the  first  remainder  will  be, 
65  I  a+  95- 
-10c       -125c 
'     -   5c" 
To  continue  the  operation,  it  is  necessary  to  take  the  second  po- 
lynomial  for  a  dividend,  and  this  remainder  for  a  divisor,  and  multi- 
ply  the  new  dividend  by  65— 10c,  or  simply  35— 5c,  since  2  is  a 
factor  of  the  first  temi  of  the  dividend.     But  we  are  not  at  liberty 
to  multiply  by  35— 5c,  if  it  is  a  factor  of  the  remainder.     There- 
fore, before  effecting  the  multiplication,  we  must  see  if  35— 5c  will 
exactly  divide  the  first  remainder  ;  we  find  that  it  does,  and  gives  for 
a  quotient  2a+35  +  c  :  whence  it  follo\vs  that  the  remainder  can  be 
put  under  the  form 

(35-5c)(2a+35+c). 


56  ALGEBRA. 

Now,  3b— 5c  is  a  factor  of  this  remainder,  and  is  not  a  factor  of 
the  new  dividend.  For,  being  independent  of  the  letter  a,  if  it  was 
a  factor  of  the  dividend  it  would  necessarily  divide  the  co- efficient 
of  this  letter  in  each  of  the  terms,  which  it  does  not ;  ^ve  may  there- 
fore  suppress  it  without  affecting  the  greatest  common  divisor. 

This  suppression  is  indispensable,  for  otherwise  a  new  factor  would 
be  introduced  into  the  dividend,  and  then  the  two  polynomials  con- 
taining a  factor  they  had  not  before,  the  greatest  common  divisor 
would  be  changed;  it  would  be  combined  with  the  factor  Sb—5c, 
which  should  not  form  a  part  of  it. 

Suppressing  this  factor,  and  effecting  the  new  division,  M'e  obtain 
an  exact  quotient ;  hence 

2a+3^+c  is  the  greatest  conomon  divisor. 
Remaek.  The  rule  for  the  greatest  common  divisor  of  two  po- 
lynomials,  may  readily  be  extended  to  three  or  more  polynomials. 
For,  having  the  polynomials  A,  B,  C,  D,  &c.  if  we  find  the  greatest 
common  divisor  of  A  and  B,  and  then  the  greatest  common  divisor 
of  this  result  and  C,  the  divisor  so  obtained  will  evidently  be  the 
greatest  common  divisor  of  A,  B  and  C  ;  and  the  same  process  may 
be  applied  to  the  remaining  polynomials. 

2.  Find  the  greatest  common  divisor  of  x*—l  and  a^^+o;'. 

Ans.     l+x"^. 

3.  Find  the  greatest  common  divisor  of  4a'  — 2a^  — 3a  +  l  and 
3a'  — 2a  — 1.  Ans.     a  —  1. 

4.  Find  the  greatest  common  divisor  of  a*—x*  and  a^—a'x—ax'' 
+x\  Ans.     a''—x\ 

5.  Find  the  greatest  common  divisor  of  SGa"- 18a^— 27a^  +  9a' 
and  27a-h''-18a'b'  —  9a'b\ 

Ans.     9aXa—l). 

6.  Find  the  greatest  common  divisor  of 

qnp'  +  2npY—^npcf—2nq''  and  2tnpY—A7np*—mp''q+2mpq\ 

Ans.    p—q. 


OF  FRACTIONS.  57 

7.  Find   the  greatest  common   divisor  of  the  two  polynomials 
l5a'+l0a'b-}-4:a'b^-\-6a'b'  —  2ab'' 
12a'b^+38a^b'+16ab*-l0b\ 

Ans.     Sa^+2ab-b\ 

CASE  11. 

71.  To  reduce  a  mixed  quantity  to  the  form  of  a  fraction. 

RULE. 

Multiply  ilie  entire  part  by  tlie  dekominator  of  ihefracUon  :  then 
connect  this  jJrodiict  with  the  terms  of  the  numerator  by  the  rules  for 
addition,  and  under  the  result  place  the  given  denominator. 


EXAMPLES. 

1. 

Reduce 

X- 

(a^-x^) 

X 

to  the  form  of  a 

fraction. 

X  — 

a'-x' 

x->-{a'-x')     2x 

X 

.     Ans. 

X 

X 

2. 

Reduce 

X- 

2a 

to  the  form  of  a 

L  fraction. 

flX'-X^ 

Ans.     . 

2a 


2x— 7 

3.  Reduce     5H — — to  the  form  of  a  fraction 

ox 


17x-7 

Ans.     — . 

3x 


a; — a — 1 

4.  Reduce     1— to  the  form  of  a  fraction. 


2a-x+l 

Ans. . 

a 


X— 3 

5.  Reduce     l+2x— r —    to  the  form  of  a  fraction. 
ox 


10x=+4x4-3 
^^^-     5^ • 


58  ALGEBRA. 

CASE  III. 

72.  To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

RULE. 

Divide  the  numerator  by  the  denominator  for  the  entire  part,  and 
place  the  remainder,  if  any,  over  the  denominator  for  the  fractional 
part. 


EXAMPLES. 

1. 

Reduce 

ax-\-a^ 

X 

to  a  mixed  quantity. 

ax+a""           ft^ 
=a+—     Ans. 

2. 

Reouce 

ax—x"^ 

X 

to  an  entire  or  mixed  quantity. 

Ans.     a—x. 

3. 

Reduce 

ah-2a^ 
b 

to  a  mixed  quantity. 

2a' 
Ans.     a——r. 

0 

4. 

Reduce 

a^-x' 
a—x 

to  an  entire  quantity. 

Ans.     a-\-x 

5. 

Reduce 

x'-y' 
x—y 

to  an  entire  quantity. 

Ans,     x'+xy+y^. 

6. 

Reduce 

10a;=-5x+3 
r-; to  a  mixed  quantity. 

3 
Ans.     2a;- 1+—. 

,^K^  CASE  IV. 

73.  To  reduce  fractions  having  different  denominators  to  equiva- 
lent fractions  having  a  common  denominator. 


OF  FRACTIONS  59 


RULE. 


Multiply  each  numenttor  into  all  the  denominators  except  its  oton, 
for  the  new  numerators,  and  all  the  denominators  together  for  a  com- 
mon denominator. 


EXAMPLES. 

a  h 

1.  Reduce     —  and  —     to  equivalent  fractions  having  a  com- 

mon  denominator. 

ay^c^ac 


,     ,      ,  „  .  the  new  numerators. 
hxl>=h^  ) 

and     .     .     hxc=bc     the  common  denominator. 


a  a-\-b 

'    ^  to  tractions,  having  a  C( 

ab+P 


2.  Reduce     -r-  and  to  fractions,  having  a  common  de- 

be  '  » 


Ans.     -r-  and 


be  be 

3x      2b 

3.  Reduce     — -,     7;—,     and  d,  to  fractions  having  a  common  dc- 

9cx         Aab  Gacd 

nominator.  Ans.     — — ,     — —  and  — — . 

Qac         Qac  (\ac 

„    ,  3       2a;  2x 

4.  Reduce     — -,     — ,  and  a-\ ,     to  fractions  having   a  com- 

9a  Qax  12a=+24x 

mon  denommator.  Ans.     ,     ,  and 

12a         12a 


12a 


r     r,    ^  la'  a'+x' 

5.  Reduce     -— ,     -—  and  ■ ,     to  fractions  having  a  com- 

2        3  a+x  ° 


mon  denominator. 


3a+3a:        2a^+2a^x  6a=+6a;' 

Ans.     - — — -,     — — — — ,  and 


Qa  +  Qx'       Qa-\-Gx   '  6a+6.T 


CASE  V. 
74.  To  add  fractional  quantities  together. 


60  ALGEBRA. 

RULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  denoyninator :  then 
add  the  numerators  togetJier  and  place  their  sum  over  the  common  rfe- 
nominator. 

EXAMPLES. 

1.  Find  the  sum  of    — ,     — ,  and  -j. 

Here,      .     axdxf=adf' 

cxixf—cbf  y  the  new  numerators. 
exbxd=ebd 

And   .     .     bxdxf=idf     the  common  denominator. 
adf       cbf        eld      adf+cbf-\-ebd 

"'"^"'  -w^W^W^  '    bdf —  '^^  '""^- 

^  Sx^       ^  ^   ^      2ax 

2.  To     a 7-  add  b-\ . 

b  c 

2abx—'6cx'' 
Ans.     a+6+- 


bc 

XXX  X 

3.  Add     — -,     —  and  —     together.  Ans.     x+—-. 

4>         o  4  12 

^     ^  ,^     x-2  4a;  ,  ^  iar-14 

4.  Add     — - —  and  -—     together.  \ns.     — . 

o  1  21 

x—2  2a,'— 3 

5.  Add     xH — —  to  ^x-\ — . 

lOx-lT 


Ans.     Ax-\- 


12 

5a;*  x-^a 

6.  It  is  required  to  add     4a;,  — - — ,  and   — - —     together. 
2a  2a;  ° 

5a;^+aa;+a' 
Ans.     4a;4— 


2aa; 


«    T    ■  ■     •■  ,,     2.r      7.T         ,     2a;+l 

7.  It  IS  required  to  add     — ,     — ,  and —     together. 

tj         4  u 

49a;+12 
Ans.     2a;+- 


60 


OF  FRACTIONS.  61 

8.  It  is  required  to  add     4x,  — ,  and  2+—     together. 

441;+ 90 


Atis      Ax-\— 


45 


9.  It  is  required  to  add     3a: +—  and  x — —     together 


2x  8x 

together. 

23a; 
Am.     2x-{ — 7=— 
45 


CASE  VI. 
75.  To  subtract  one  fractional  quantity  from  another. 

RULE. 

I.  Reduce  the  fractions  to  a  common  denominator. 

II.  Subtract  the  numerator  of  the  fraction  to  he  subtracted  f'om  the 
numerator  of  the  other  fraction,  and  place  the  difference  over  tJie  coip~ 
mon  denominator. 

EXAMPLES. 

X  *~-  Qi  2rt  ^—  4^ 

1.  Find  the  difference  of  the  fractions       ^,     and  — 

2b  3c 

Here,  (x—  a)x  3c=:3cx— 3ac  ,     , 

^  ^  ^    the  numerators 


(2a— 4a;)x2J=4a3— 85a; 
And,  25  x3c=65c         the  common  denominator. 

^cx — Zac      4ab—8bx      3cx—Sac  —  4:ab-\-8bx 

"^"'^^'   —Wc 6br-= eTc -•    '^"^• 


12.r  3.r 

2.   Required  the  difference  of    — —  and  — 


3.  Required  the  difference  of  5i/  and     — . 


39a; 
Ans.     -3^. 


Ans.     —- — , 


62 


Sx  2x 

4.  Required  the  difference  of    —  and  — . 

^"^-     -63- 

5.  Requia-ed  the  difference  between    — ^ —  and  —-. 

0  a 

dx-\-ad—bc 
Alls. 


bd 


Sx+a  2x+l 

6.  Required  the  difference  of    — ^ —  and 


Ans 


bb  8 

24a;+8a— lOZ'a;  — 355 


40^> 

X                   X — a 
7.   Required  the  difference  of    3a;+—  and  x . 

cx-\-bx—ab 


Ans.     2x+- 


bc 

CASE  VII. 
76,  To  multiply  fractional  quantities  together. 
RULE. 

If  the  quantities  to  he  multiplied  are  mixed,  reduce  them  to  a  frac 
tionalform  ;  then  multiply  the  numerators  together  for  a  numerator 
and  the  denominators  together  for  a  denominator. 


EXAMPLES. 

1. 

Multiply 

bx 

-y-a- 

bx 

a^+l 
a 

X 

en 

3e,  .     .     . 

•     • 

a^+bx 

X 
a 

c 
1^' 

a^c-{-bcx 
ad 

2. 

Required 

3a; 

the  product  of    — - 

and 

Sa 
~b' 

Ans. 


9ax 


OF  FRACTIONS.  63 

2a;  Sx' 

3.  Required  the  product  of    —  and   — . 

Ans.     -^— . 
5a 

2a;  2ab  Sac 

4.  Find  the  continued  product  of  — ,  ,  and  ^,  . 

*  a         c  2b 

Alls.     9ax. 

hx  a 

5.  It  is  required  to  find  the  product  of    b-\- —  and  — . 

ab-{-bx 


Ans. 

X 

x'^  —  P  x'^+b' 

6.  Required  the  product  of    — r and  — y— — . 

x'—b- 
Ans. 


x-}-l  x—1 

7.  Required  the  product  of    x-i ,  and         , 


Ans. 


b\+bc'' 
ax"—ax-\-x'^—\ 


a--\-ab 


d^  0^ y^ 

8.  Required  the  product  of    a-\ ■  by 


a—x  a;  -\-x' 


Ans. 


ax-{-ax'—x^—x* 

CASE  VIII. 

77.  To  divide  one  fractional  quantity  by  another. 

RULE. 

Reduce  the  mixed  quantities,  if  there  are  any,  to  a  fractional  form : 
then  invert  the  terms  of  the  divisor  and  multiply  the  fractions  together 
as  in  the  last  case. 

EXAMPLES. 

1.  Divide     ....    a——-  by  — . 
2c  g 

b  _2ac—b  . 

^~2^~~2r~ 


64 

ALGEBRA. 

Hence, 

b    .  /      2ac-b      g 
"     2c-  g-      2c     ^f 

2acg- 
-       2c/ 

:^. 

Ans. 

2.  Let 

7x                            12 
—   be  divided  by  — . 

O                                        lo 

^MS. 

91a; 
60  * 

3.  Let 

4a-^ 
— -—  be  divided  by  5x. 

4x 

4.  Let 

— - —  be  aivided  by  -— . 
b                                    6 

^ns. 

a:+l 
4a;  * 

5.  Let 

X                                       x 

be  divided  by  -— . 

X — 1                               2 

^?JS. 

2 

X-\' 

6.  Let 

—  be  divided  by  —^. 

Ans. 

bbx 
2a  ' 

7.  Let 

-r—-r  be  divided  by      ,  ,  . 
Serf                         •'      4(Z 

Ans. 

x-b 
6c'x' 

8.  Let 

..■-2te+*'   "'"'■"■ledby- 

Ans. 

78.  We  will  add  but  a  single  proposition  more  on  the  subject  of 
fractions.     It  is  this. 

If  the  same  number  be  added  to  each  of  the  terms  of  a  projier  fraction, 
the  new  fraction  resulting  from  this  addition  will  be  greater  than  the 
first ;  but  if  it  be  added  to  the  terms  of  an  hwproper  fraction,  the  re- 
sulting fraction  will  be  less  than  the  first. 

a 
Let  the  fraction  be  expressed  by     — ,  and  suppose  a<Cb. 

Let   m    represent  the  number    to  be  added  to  the  terms  :    the 

a-\-m 

fraction  then    becomes . 

b-\-m 

In  order  to  compare  the  two  fractions,  they  must  be  reduced  to  the 

,  .  ,  •  ,      •  ab+am     ^      ,     ^  ,     ab+bm 

same  denominator,  which  eives     -7- — ; —     for  the  first,  and    -^ 

b^+bm  b^'  +  bm 

for  the  second. 


EQUATIONS  OF  THE  FIRST  DEGREE.  65 

Now,  the  denominators  being  the  same,  that  fraction  will  be  the 
greatest  which  has  the  greater  numerator.  But  the  two  numera- 
tors, ah-\-am,,  and  ai-\-bm,  have  a  common  part  ab  ;  and  the  part  bm 
of  the  second  is  greater  than  the  part  a/n  of  the  first,  since  J>a. 
Hence  the  second  fraction  is  greater  than  the  first. 

If  the  given  fraction  is  improper,  or  a>5,  it  is  plain  that  the  nu- 
merator of  the  second  fraction  will  be  less  than  that  of  the  first, 
once  bm  would  be  less  than  am. 


CHAPTER  n. 
Of  Equations  of  the  First  Degree. 

79.  An  Equation  is  the  expression  of  two  equal  quantities  with 
the  sign  of  equality  placed  between  them.  Thus,  x^=a-{-b  is  an 
equation,  in  which  x  is  equal  to  the  sum  of  a  and  b, 

80.  By  the  definition,  every  equation  is  composed  of  two  parts, 
separated  from  each  other  by  the  sign  =.  The  part  on  the  left  of 
the  sign,  is  called  the  first  member,  and  the  part  on  the  right,  is  called 
the  second  member ;  and  each  member  may  be  composed  of  one  or 
more  terms. 

81.  Every  equation  may  be  regarded  as  the  enunciation,  in  alge- 
braic language,  of  a  particular  problem.  Thus,  the  equation 
a;+a;=:  30,  is  the  algebraic  enunciation  of  the  following  problem ; 

To  find  a  number  which,  being  added  to  itself,  shall  give  a  sum 
equal  to  20. 

Were  it  required  to  solve  this  problem  we  should  first  express  it 
in  algebraic  language,  which  would  give  the  equation 

x-\-  a;=:30. 

By  adding  a;  to  itself,  we  have 2a;=:30. 

and  by  dividing  by  2,  we  obtain        ....  x=  15. 

6* 


66  ALGEBRA. 

Hence  we  see  that  the  solution  of  a  problem  by  algebra,  consists 
of  two  distinct  parts. 

15^.  To  express  algebraically  tlie  relation  between  the  known  and 
uiiknown  quantities. 

2d.  To  find  a  value  for  the  unknown  quantity,  in  terms  of  those 
which  are  known,  which  substituted  in  its  place  in  the  given  equation 
vnll  satisfy  the  equation ;  that  is,  render  tJie  first  meinber  equal  to  the 
second. 

This  latter  part  is  called  the  solution  of  the  equation. 

82.  An  equation  is  said  to  be  verified,  when  such  a  value  is  sub- 
stituted  for  the  unknown  quantity  as  will  prove  the  two  members  of 
the  equation  to  be  equal  to  each  other. 

83.  Equations  are  divided  into  different  classes.  Those  which 
contain  only  the  first  power  of  the  unknown  quantity,  are  called 
equations  of  the  first  degree.     Thus, 

ax  -\-  b  =1  cx+d        is  an  equation  of  the  1st.  degree. 
2x^—'6x  =5  —2a;''     is  an  equation  of  the  2d.  degree. 
4a;'  — 5a;'' =3x4- 11      is  an  equation  of  the  3d.  degree. 
In  general,  the  degree  of  an  equation  is  denoted  by  the  greatest 
of  the  exponents  with  which  the  unknown  quantity  is  affected. 

84.  Equations  are  also  distinguished  as  numerical  equations  and 
literal  equations.  The  first  are  those  which  contain  numbers  only, 
with  the  exception  of  the  unknown  quantity,  which  is  always  de- 
noted  by  a  letter.  Thus,  4a;— 3=2a;-f  5,  3a;'  — a;=8,  are  numerical 
equations.  They  are  the  algebraical  translation  of  problems,  in 
which  the  known  quantities  are  particular  numbers. 

The  equations  ax—b=:zcx-\-d,  ax^-{-bx-=c,  are  literal  equations, 
in  which  the  given  quantities  of  the  problem  are  represented  by 
letters. 

85.  It  frequently  occurs  in  algebra,  that  the  algebraic  sign  +  or 
— ,  which  is  written,  is  not  the  true  sign  of  the  term  before  which 
it  is  placed.  Thus,  if  it  were  required  to  subtract  —b  from  a,  we 
should  write 


EQUATIONS  OF  THE  FIRST  DEGREE.  67 

Here  the  true  sign  of  the  second  term  of  the  binomial  is  pkis,  al- 
though  its  algebraic  sign,  which  is  written  in  the  first  member  of  the 
equation,  is  — .  This  minus  sign,  operating  upon  the  sign  of  h, 
which  is  also  negative,  produces  a  plus  sign  for  b  in  the  result. 
The  sign  which  results,  after  combining  the  algebraic  sign  with  the 
sign  of  the  quantity,  is  called  the  essential  sign  of  the  term,  and  is 
often  different  from  the  algebraic  sign. 

By  considering  the  nature  of  an  equation,  we  perceive  that  it 
must  possess  the  three  following  properties. 

1st.  The  two  members  are  composed  of  qu  antities  of  the  same  kind. 

2d.    The  two  members  are  equal  to  each  other. 

3d.    The  essential  sign  of  the  two  members  must  be  the  same. 

Equations  of  the  First  Degree  involving  but  one  unknown 
quantity. 

86.  An  axiom  is  a  self-evident  proposition.  We  may  here  state 
the  following. 

1.  If  equal  quantities  be  added  to  both  members  of  an  equation, 
the  equality  of  the  members  will  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  from  both  members  of  an 
equation,  the  equahty  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiphed  by  the  same 
number,  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same  num. 
bar,  the  equality  will  not  be  destroyed. 

87.  The  transformation  of  an  equation  consists  in  changing  its 
form  without  affecting  the  equality  of  its  members. 

The  following  transformations  are  of  continued  use  in  the  resolu- 
tion  of  equations. 

First  Transformation. 

88.  When  some  of  the  terms  of  an  equation  are  fractional,  to  re- 
duce the  equation  to  one  in  which  the  terms  shall  be  entire.    • 


68 

Take  the  equation, 


2x     3         X 


First,  reduce  all  the  fractions  to  the  same  denominator,  by  the 
knowTi  rule  ;  the  equation  becomes 

48a;       54a;       12a; 
"72         72""*"   72  ~^^ 
and  since  we  can  multiply  both  inembers  by  the  same  number  with- 
out  destroying  the  equality,  we  will  multiply  them  by  72,  which  is 
the  same  as  suppressing  the  denominator  72,  in  the  fractional  terms, 
and  multiplying  the  entire  term  by  72  ;  the  equation  then  becomes 

48x'— 54a;+12a;=792. 
or  dividing  by  6  8a;—   9a;+  2a;=:132. 

89.  The  last  equation  could  have  been  found  in  another  manner 
by  employing  the  least  common  multiple  of  the  denominators. 

The  common  multiple  of  two  or  more  numbers  is  any  number 
which  they  will  both  divide  without  a  remainder ;  and  the  least 
common  multiple,  is  the  least  number  which  they  will  so  divide. 
The  least  common  multiple  will  be  the  pz'oduct  of  all  the  numbers, 
when,  in  comparing  either  with  the  others,  we  find  no  common  fac- 
tors. But  when  there  are  common  factors,  the  least  common  mul- 
tiple will  be  the  product  of  all  the  numbers  divided  by  the  product 
of  the  common  factors. 

The  least  common  multiple,  when  the  numbers  are  small,  can 

generally  be  found  by  inspection.     Thus,  24  is  the  least  common 

multiple  of  4,  6,  and  8,  and  12  is  the  least  common  multiple  of 

3,  4  and  6. 

2a;      3         a; 
Take  the  last  equation     — — a;+— =11. 

We  see  that  12  is  the  least  common  multiple  of  the  denomina- 
tors, and  if  we  multiply  all  the  terms  of  the  equation  by  12,  and 
divide  by  the  denominators,  we  obtain 

8a;— 9.r+2x=132. 
the  same  equation  as  before  found. 


EQUATIONS  OF  THE  FIRST  DEGREE.  69 

90.  Hence,  to  make  the  denominators  disappear  from  an  equation, 
we  have  the  following 

RULE. 

I.  Form  the  least  common  multiple  of  all  the  denominators. 

II.  Multiply  each  of  the  entire  terms  by  this  multiple,  and  each  of 
the  fractional  terms  hy  the  quotient  of  this  multiple  divided  by  the  de- 
nominator  of  the  term  thus  multiplied,  and  omit  the  denominators  of 
the  fractional  terms. 

EXAMPLES. 

1.  Clear  the  equation     ~r-\—^ — 4=3     of  its  denominators. 


Ans.     7x+5x-140=105. 


2.  Clear  the  equation     -7 —-\-fz=g. 


Alls.     ad—bc-\-ldf=zbdg. 


3..  In  the  equation 


ax      2c^x  Abc'^x      5a^        2c^ 

b        ab  a?  W  a 

the  least  common  multiple  of  the  denominators  is  (V'V  ;  hence  clear- 
ing the  fractions,  we  obtain 

a''bx-2a%c''x^\aW=M''&x-ha^-^2a'^We—2,aW. 

Second  Ti'ansformatiun. 

91.  When  the  two  members  of  an  equation  are  entire  polynomials, 
to  transpose  certain  terms  from  one  member  to  the  other. 

Take  for  example  the  equation      ....     5x  — 6=84-2x. 

If,  in  the  first  place  we  subtract  2a?  from 
both  members,  the  equality  will  not  be  de- 
stroyed, aud  we  have     5a;— 6  — 2a;=:8. 

Whence  we  see  that  the  term  2a;,  which  was  additive  in  the 
second  member  becomes  subtractive  in  the  first. 


'!0  ALGEBRA. 

In  the  second  place  if  we  add  6  to  both 
members,  the  equahty  will  still  exist  and 

we  have 5x— 6  — 2a;  +  6=8+6. 

Or,  since  —6  and  +6  destroy  each  other     5a;— 2a;=8+6. 

Hence  the  term  which  was  subtractive  in  the  first  member,  passes 
into  the  second  member  with  the  sign  of  addition. 

Again,  take  the  equation        ax-\-b=d—cx. 

If  we  add  ex  to  both  members 
and  subtract  b  from  them,   the 

equation  becomes        ....     ax-\-b-\-cx—b=d—cx+cx—b. 
or  reducing ax-\-cx=d—b. 

Therefore,    for   the    transposition   of  the  terms,   we   have    the 
following 

RULE. 

Any  term  of  an  equation  may  be  transposed  from  one  member  to  the 
other  by  changing  its  sign. 

92.  We  will  now  apply  the  preceding  principles  to  the  resolution 

of  the  equation, 

4a;— n=2,r+5. 

by  transposing  the  terms  —  3  and  2a;  it  becomes 

4a;— 2a;=54-3 

Or  reducing        .  2x=;8 

8 
Dividing  by  2     .  a;=— =4. 

Now,  if  4  be  substituted  in  the  place  of  x  in  the  first  equation,  it 
becomes 

4x4—3=2x4+5 
or      ....  13=13. 

Hence,  the  value  of  x  is  verified  by  substituting  it  for  the  unknown 
quantity  in  the  given  equation. 

For  a  second  example,  take  the  equation 
5a;     Ax  7       13a; 

1  o 

12      3  8         6    • 


EQUATIONS  OF  THE  FIRST  DEGREfc.  71 

By  making  the  denominators  disappear,  we  have 

lOx  — 32j;— 312=  21  — 52x 

or,  by  transposing       .      lOx'— 32a;+52a;=  21  +312 

by  reducing      .     .     .  300;=  333 

333      111 
dividing  by  30       .     .  a,'=-— -=:— — — =11,1. 

oO         10 

a  result  which  may  be  verified  by  substituting  it  for  x  in  the  given 
equation. 

For  a  third  example  let  us  take  the  equation 
(3a— ic)  (a—b)+2ax=4:h{x-[-a). 
It  is  first  necessary  to  perform  the  multiplications  indicated,  in  or- 
der to  reduce  the  two  members  to  two  polynomials,  and  thus  be  able 
to  disengage  the  unknown  quantity  x,  from  the  known  quantities. 
Having  done  that,  the  equation  becomes, 

Sa''  —  ax—2ab-\-bx-\-2ax=z4:bx-\-4:ab. 
or  by  transposing      .         —  cw;+Z'a;+2avr— 4ix  =4a5+3ai— 3a' 
by  reducing       .  .  ax—Sbx  =lab— So' 

Or,  (Art.  48).        .  .  {a-Sb)x=:7ab-Sa^ 

Dividing  both  members  by  a—Zb  we  find 
7a5— 3a^ 
^=     a-3b    ' 
93.  Hence,  in  order  to  resolve  any  equation  of  the  first  degree, 
we  have  the  following  general 

RULE. 

I.  If  there  are  any  denominators,  cause  them  to  disappear,  and  per- 
form, in  both  members,  all  the  algebraic  operations  indicated  :  we  thus 

obtain  an  equation  the  tico  members  of  which  are  entire  polynomials. 

II.  Tlien  transpose  all  the  terms  affected  with  the  unknown  quantity 
into  the  first  member,  and  all  the  known  terms  into  the  second  member. 

III.  Reduce  to  a  single  term  all  the  terms  involving  x :  this  term 
will  be  composed  of  two  factors,  one  of  which  will  be  x,  and  the  other 
all  the  multipliers  of  x,  connected  with  their  respective  signs. 


72  ALGEBRA. 


IV.  Divide  both  members  by  the  number  or  polynomial  by  which  the 
unknown  quantity  is  multiplied. 


EXAMPLES. 

1.  Given  3a;— 2+24=31  to  find  X.  Ans.     x=S. 

1 

2.  Given  a;+ 18= 3a;— 5  to  find  a;.  Ans.     a;=ll— . 

3.  Given  6  — 2a;+10=20  — 3a;— 2  to  find  x.  Ans.     x=2. 

4.  Given  x+— a;+— a;=ll  to  find  a;.  Ans.     x=6. 

Z  o 

1  6 

5.  Given  2a; — — a;+l  =  5a;— 2  to  find  x.  Ans.     x=—, 

a 

6.  Given    Saj:+— — S=bx—a  to  find  x. 

6-Sa 

Atis.     X-. 


'6a-2b 


7.   Given  — - — 1-^=20 —   to  find  a;. 


1 

Ans.     a;=23— . 
4 


a;+3      X  x—b 

8.  Given  ~^+"§"^'*'"~4~  ^°  ^' 


Ans.     a;=3— . 
lo 


ax—b       a       bx       bx—a 
9.  Given  — - — +y  =  y 3—     ^°  ^"'^  ^• 


3* 

Ans.     X- 


'2a  — 2b' 
10.  Find  the  value  of  x  in  the  equation 

a— A  a+o  0 

a*  +  Sa^b+'ia''b''  —  6aP  +  2b* 


Ans.     x= 


2b(2a''+ab—b') 


EQUATIONS  OF  THE  FIRST  DEGREE.  73 

Of  Questions  producing  Equations  of  the-  First  Degree 
involving  hut  a  single  unknown  que 


94.  It  has  already  been  observed  (Art.  81),  that  the  sohition  of 
a  problem  by  algebra,  consists  of  two  distinct  parts. 

1st.  To  express  the  conditions  of  the  problem  algebraically  ; 
and 

2d.  To  disengage  the  unknown  from  the  known  quantities. 

We  have  already  explained  the  manner  of  finding  the  value  of 
the  unknown  quantity,  after  the  question  has  been  stated ;  and  it 
only  remains  to  point  out  the  best  methods  of  enunciating  a  problem 
in  the  language  of  algebra. 

This  part  of  the  algebraic  resolution  of  a  problem,  cannot,  like 
the  second,  be  subjected  to  any  well  defined  rule.  Sometimes  the 
enunciation  of  the  problem  furnishes  the  equation  immediately  ;  and 
sometimes  it  is  necessary  to  discover,  from  the  enunciation,  new  con- 
ditions  from  which  an  equation  may  be  formed.  The  conditions 
enunciated  are  called  explicit  conditions,  and  those  which  are  de- 
duced from  them,  implicit  conditions. 

In  almost  all  cases,  however,  we  ai-e  enabled  to  discover  the  equa- 
tion  by  applying  the  following 

RULE. 

Consider  the  problem  solved ;  and  then  indicate,  hj  means  of  alge. 
hraic  signs,  upon  the  known  and  unknown  quantities,  the  same  course 
of  reasoning  and  operations  which  it  ivould  he  necessary  to  perform-, 
in  order  to  verify  the  unknown  quantity,  had  it  been  given. 

QUESTIONS. 

1.  Find  a  number  such,  that  the  sum  of  one  half,  one  third,  and 
one  fourth  of  it,  augmented  by  45,  shall  be  equal  to  448. 

Let  the  required  number  be  denoted  by     .         .         .  x  , 

7 


74  A.LGEBRA. 

X 

Then,  one  half  of  it  will  be  denoted  by     .         .         .        — . 

X 

one  third  of  it  .         .         by     .         .         •         -^. 

o 

X 

one  fourth  of  it       .         .         by     .         .         .         — . 

And  by  the  conditions,     —-+— +-^+45=448. 
^        o        4 

Or  by  subtracting  45  from  both  members, 

XXX 

_+_+_=403. 

B^  clearing  the  terms  of  their  denominators,we  obtain 

6a;+4a;+3a;=4836. 

or         .         .  13a;=:4836'. 

4836 
Hence  .  0;=— — -=372. 

lo 

Let  this  result  be  verified. 

^72        S72        S72 

-^-+-— +-^-l-45=-186  +  124  +  93  +  45=448, 
2  o  4 

2.  What  number  is  that  whose  third  part  exceeds  its  fourth,  by 

16. 

Let  the  required  number  be  represented  by  x.     Then, 


-3-^= 

=     the  third  part. 

1 

the  fourth  part 

And  by  the  question 

1         1 
-x--.=  16. 

or,      . 

4a;-3a:=192. 
a;=192. 

Verification. 

192 

192 

-^ T-=64-48=16. 

3  4 


EQUATIONS  OF  THE  FIRST  DEGREE.  75 

3,  Divide  $1000  between  A,  B,  and  C,  so  that  A  shall  have  i72 
more  than  B,  and  C  $100  more  than  A. 


Let       .     . 

x=               B's  share  of  the  $1000. 

Then     .     . 

x-\-  12=     A's  share. 

And      .     . 

x+n2=     C's  share. 

Their  sum 

3x4-244=1000. 

Whence, 

3a;=  1000-244=756 

or 

756 

a;=— ;— -=$252=     B's  share. 
3 

X-}-  72=252+  72=$324=     A's  share. 
And  a;+172=252  +  172=$424=     C's  share. 

Verification. 
252+324+424=1000. 
4.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third  part, 
21  gallons  were  afterwards  drawn,  and  the  cask  being  then  gauged, 
appeared  to  be  half  full :  how  much  did  it  hold  ? 
Suppose  the  cask  to  have  held  x  gallons. 

a; 
Then,         — =     what  leaked  away. 


And  —  +  21=     all  that  was  taken  out  of  it. 

o 

x  1 

Hence,       — +  21=— a;     by  the  question. 

o  Z 

or  2a;+126=3x-. 

or  —  .-c  =  —  126. 

or  X  =     126,     by  changing  the  signs  of  both 

members,  which  does  not  destroy  their  equahty. 


Verification. 


126  126 

__+21=42+21=63=-^. 

5.  A  fish  was  caught  whose  tail  weighed  9lb. ;  his  head  weighed 


76  ALGEBRA. 

as  much  as  his  tail  and  half  his  body,  and  his  body  weighed  as 
much  as  his  head  and  tail  together  ;  M'hat  was  the  weight  of  the 
fish? 

Let  .  2  a;=     the  weight  of  the  body. 

Then        .  9+a;=     weight  of  the  head. 

And  since  the  body  weighed  as  much  as  both  head  and  tail 
2a;=9+   9+a; 
or       .         .     2x— a;=:18     and     a;=18. 

Veiification. 

2a;=:36Z^'=     weight  of  the  body. 
9+a;=27/3=     weight  of  the  head. 
9Zi=     weight  of  the  tail. 
Hence,      .  72  lb=     weight  of  the  fish. 

6.  A  person  engaged  a  workman  for  48  days.  For  each  day 
that  he  laboured  he  received  24  cents,  and  for  each  day  that  he  was 
idle,  he  paid  12  cents  for  his  board.  At  the  end  of  the  48  days,  the 
account  was  settled,  when  the  labourer  received  504  cents.  Re. 
quired  the  number  of  working  days,  and  the  number  of  days  lie  was 
idle. 

If  these  two  numbers  were  known,  by  multiplying  them  respec- 
tively by  24  and  12,  then  subtractmg  the  last  product  from  the  first, 
the  result  would  be  504.  Let  us  indicate  these  operations  by  means 
of  algebraic  signs. 

Let     .     .         X  =     the  number  of  working  days. 

48— a;  =     the  number  of  idle  days. 
Then         24X'i-'  =     the  amount  earned,  and 

12(48— x)=     the  amount  paid  for  his  board. 
Then  24a;— 12(48— x)    =504     what  he  received, 

or  24x-576  +  12x=504. 

or  36a;=504+576  =  1080 

1080 
and  a;=— ^j^=30     the  working  days. 

whence,  48—30=18     the  idle  days. 


EQUATIONS  OF  THE  FIRST  DEGREE.  77 

Verification. 

Thirty  day's  labor,  at  24  cents  a  day, 

amounts  to 30x24=720  cts. 

And  18  day's  board,  at  12  cents  a  day, 

amounts  to 18x12=216  cts. 

And    720—216=504,  the  amount  received. 
This  question  may  be  made  general,  by  deno- 
ting the  whole  number  of  working  and  idle  days. 
The  amount  received,  for  each  day  he  worked, 
The  amount  paid  for  his  board,  for  each  idle 

day, 

And  the  balance  due  the  laborer,  or  the  result 

of  the  account, 

As  before,  let  the  number  of  working  days  be 
represented      ....... 

The  number  of  idle  days  will  be  expressed 
Hence,  what  he  earns  will  be  expressed 
and  the  sum  to  be  deducted,  on  account  of  his  board. 
The  equation  of  the  problem  therefore  is, 
ax — h(^n  —  x)^c 
whence  ax—h  n+lx=c 

{a-\-h)x=c  +^n 
c  -{-hn 

c  -\-hn      nn-\-ln—c  —  hn 


by 

n. 

by 

a. 

by 

h. 

by 

c. 

by 

X. 

by 

n—x. 

by 

ax. 

by 

b{n--x). 

and  consequently, 


a  +b  a-i-b 

071  — c 


a+b 

7.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  60  leaps.  He 
makes  9  leaps  while  the  greyhound  makes  but  6  ;  but  three  leaps  of 
the  greyhound  are  equivalent  to  7  of  the  fox.  How  many  leaps 
must  the  greyhound  make  to  overtake  the  fox  ? 

From  the  enunciation,  it  is  evident  that  the  distance  to  be  passed 


78  ALGEBRA. 

over  by  the  greyhound  is  composed  of  the  60  leaps  which  the  fox  is 
in  advance,  plus  the  distance  that  the  fox  passes  over  from  the  mo- 
ment when  the  greyhound  starts  in  pursuit  of  him.  Hence,  if  we 
can  find  the  expression  for  these  two  distances,  it  will  be  easy  to 
form  the  equation  of  the  problem. 

Let  «=:  the  number  of  leaps  made  by  the  greyhound  before 
he  overtakes  the  fox. 

Now,  since  the  fox  makes  9  leaps  while  the   greyhound  makes 

9  3 

but  6,  the  fox  will  make     —  or  —     leaps   while  the   greyhound 

makes  1 ;  and,  therefore,  while  the  greyhound  makes  x  leaps,  the 

3 

fox  will  make     —x     leaps. 

Hence,  the  distance  which  the  greyhound  must  pass  over,  will  be 

3 

expressed  by     60-\-—x     leaps  of  the  fox. 

It  might  be  supposed,that  in  order  to  obtam  the  equation,  it  would 

3 

be  sufficient  to  place  x  equal  to     60+- ^a;;     but   in   doing   so,    a 

manifest  error  would  be  committed ;  for  the  leaps  of  the  greyhound 

are  greater  than  those  of  the  fox,  and  we  would  then  equate  hetero- 

geneous  numbers,   that   is,   numbers   referred   to   different  units. 

Hence  it  is  necessary  to  express  the  leaps  of  the  fox  by  means  of 

those  of  the  greyhound,  or  reciprocally.     Now,  according  to  the 

enunciation,  3  leaps  of  the  greyhound  are  equivalent  to  7  leaps  of 

7 
the  fox,  then  1  leap  of  the  greyhound  is  equivalent  to    —    leaps  of 

the  fox,  and  consequently  a;  leaps  of  the  greyhound  are  equivalent 

Ix 
to       -r-     of  the  fox. 
<i 

Ix  3 

Hence,  we  have  the  equation        — =  60+— x; 

making  the  denominators  disappear     14x=360+  9a:, 

Whence         ....      5x=360  and  x=72. 


EQUATIONS  OF  THE   FIRST  DEGREE  79 

Therefore,  the  greyhound  will  make  72  leaps  to  overtake  the  fox, 

3 
and  during  this  time  the  fox  will  make     72  X"^     or  108. 

Verification. 

72x7 
The  72  leaps  of  the  greyhound  are  equivalent  to     — r — =168 

leaps  of  the  fox. 

And  60  +  108=168,  the  leaps  which  the  fox  made  from  the 
beginning. 

8.  A  father  who  had  three  children,  ordered  in  his  will,  that  his 
property  should  be  divided  amongst  them  in  the  following  manner  : 
the  first  to  have  a  sum  a,  plus  the  nth  part  of  what  remained  after 
subtracting  a  from  the  whole  estate  ;  the  second,  a  sum  2a  plus  the 
nth  part  of  what  remained  after  subtracting  from  it  the  first  part  and 
2a  ;  the  third,  to  have  a  sum  2a  plus  the  nth  part  of  what  remain- 
ed  after  subtracting  from  it  the  first  two  parts  and  3a.  In  this  man- 
ner his  property  was  entirely  divided  ;  required  the  amount  of  it. 

Let  X  denote  the  property  of  the  father.  If  by  means  of  this 
quantity,  algebraic  expressions  can  be  formed  for  the  three  parts, 
we  may  subtract  their  sum  from  the  whole  property  x,  and  the  re- 
mainder placed  equal  to  zero,  will  give  the  equation  of  the  prob- 
lem. We  will  then  endeavour  to  determine  successively  these 
three  parts. 

Since  x  denotes  the  property  of  the  father,  x—a  is  what  remains 

after  having  subtracted  a  from  it ;  therefore  x—a  is  the  first  re- 

x—a 
mainder,  and  the  part  which  the  first  child  is  to  have,  is     a-\ > 

or  reducing  to  a  common  denominator, 

a7i-\-x—a 
=lst  part. 

In  order  to  form  the  2d  part,  this  first  part  and  2a  must  be  subtract- 

(an-\-x—a)  ,     . 

ed  from  x :  this  gives    x—2a -,  or  reducing  to  a  com- 


80  ALGEBRA. 

mon  denominator  and  subtracting, 

nx — ^an — x-\-a 


2d.  remainder. 


N  ow,  the  second  part  is  composed  of  2a,  plus  the  nih.  part  of  this 

nx—^an—x-\-a 
remainder;  therefore,  it  is     2a-\ ,     or  reducing  to 

a  common  denominator, 

2an^-\-nx—^an—x-{-a 
—^ =2d.  part. 

Subtracting  the  two  first  parts  pkis  3fl,  from  x,  we  have 
{an+x—a)      (2avF-\-nx—^an—x-\-a) 
n  ir  ' 

Or,  reducing  to  a  common  denominator,  and  performing  the  opera- 
tions indicated, 

n^x—Qarv' —^nx-^-^an-^x—a 
7. 3d.  remainder. 

„           ,        ^          .              n^x—6a7r  —  27ix-\-4.a7i-\-x—a 
Hence  the  3d  part  is     3a-] 


Or,  reducing  to  a  common  denominator, 

tian^-\-n'^x—6an^—2nx-\-4ian-\-x—a 


:3d  part. 


But  from  the  enunciation,  the  estate  of  the  father  is  found  to  be 
entirely  divided.     Hence,  the  difference  between  x,  and  the  sum  of 
the  three  parts  should  be  equal  to  zero.     This  gives  the  equation 
an  4-x— a      2an^+  nx—San—x-\-a 

,  =0. 
2,a')^-\-ivx   —%aT?—2nx-\-^:an-\-x—a 

by  making  the  denominators  disappear,  and  performing  the  opera- 
tions indicated,  we  have 

n='x— 6a7i3— S/i^'a'+lOan^+Snx— 5an— a;+a=0. 


EQUATIONS  OF  THE  FIRST  DEGREE.  81 

Whence, 

_6an'-10«n2+5an-a_a(6n='— 1071^  +  571-1) 
^^  iv'—Sn^  +  dn  —  l  ~  rv'—Sn^-i-Sn—l 
A  more  simple  equation  and  result  may  be  obtained,  by  observ- 
ing,  that  the  part  which  goes  to  the  third  child  is  composed  of  3a, 
plus  the  7ith  part  of  what  remains,  and  that  the  estate  is  then  entirely 
exhausted  ;  that  is,  the  third  child  has  only  the  sum  3a,  and  the  re- 
mainder just  mentioned  is  nothing. 

Now  the  expression  for  this  remainder  has  been  found  to  be 
ri^x—6an^—2nx-\-4:an-\-x—a 
n^  • 

Placing  this  equal  to  zero,  and  making  the  denominator  disappear, 
we  have 

n^x—6an^—2nx-\-4ia7i-\-x—a^0. 

6an^—4:an-{-a      a(6n^—4:n-\-l) 

Whence      x= t, — - — — — = ^ — „     ,  , 

7r— 2/t+l  7r— 2/1+1 

Ve}ificatio7i. 

To  prove  the  numerical  identity  of  this  expression  with  the  pre- 
ceding,  it  is  only  necessary  to  show  that  the  second  can  be  deduced 
from  the  first,  by  suppressing  a  factor  common  to  its  numerator  and 
denominator.  Now  if  we  apply  the  rule  for  finding  the  greatest 
common  divisor  (Art.  70.),  to  the  two  polynomials 

a(67i2- 1071^+571-- 1)     and     v? —Zii' J^^n—X, 
it  will  be  seen  that   ?i— 1  is  a  common  factor,  and  by  dividing  the 
numerator  and  denominator  of  the  first  expression  by  this  factor, 
the  result  will  be  the  second. 

This  problem  shows  the  beginner  how  important  it  is  to  seize 
upon  every  circumstance  in  the  enunciation  of  a  question,  which 
may  facilitate  the  formation  of  the  equation,  otherwise  he  runs  the 
risk  of  arriving  at  results  more  complicated  than  the  nature  of  the 
the  question  requires. 

The  conditions  which  have  served  to  form  successively  the  ex- 


82  ALGEBRA. 

pressions  for  the  three  parts,  are  the  explicit  conditions  of  the  pro- 
blem ;  and  the  condition  which  has  served  to  determine  the  most 
simple  equation  of  the  problem,  is  an  implicit  condition,  which  a 
little  attention  has  sufficed  to  show,  was  comprehended  in  the  enun- 
elation. 

To  obtain  the  values  of  the  three  parts,  it  is  only  necessary  to 
substitute  for  x  its  value  in  the  three  expressions  obtained  for  these 
parts. 

a(6n2— 471+1)  .     ,  , 

Ai)p]y  the  formula     x=-^ — ^     ,  ,       to  a  particular  example. 

Let  a=10000,     n=5. 

We  have 
10000(6x25-4x5  +  1)     10000x131      1310000 


25-10  +  1  16  16 

To  verify  the  enunciation  in  this  case  : 


=  81875. 


81875-10000 

The  first  child  should  have,    lOOOOH r ,     or  24375. 

o 

There  remains  then  81875  — 24375,  or  57500,  to  divide  between 

the  other  two  children. 

57500-20000 

The  second  should  have,    20000  H z ,  or   27500. 

o 

Then  there  remains  57500  —  27500,  or  30000,  for  the  third 
child.  Now  30000  is  triple  of  10000  ;  hence  the  problem  is  verified. 

We  can  give  a  more  simple  and  elegant  solution  to  this  problem, 
but  it  is  less  direct.  It  also  depends  upon  the  remark,  that  after  hav- 
ing  subtracted  3a  and  the  two  first  parts  from  the  whole  estate,  no- 
thing  remains. 

Denote  the  three  remainders  mentioned  in  the  enunciation  by  r, 

r',  r" .     The  algebraic  expressions  for  the  three  parts  will  be 

r  r'  r" 

fl+— ,     2«+— ,     3a+— . 
n  n  n 

Now,  1st.  From  the  enunciation,  it  is  evident  that  r"=0. 

Therefore  the  third  part  is  3a. 


EQUATIONS  OF  THE  FIRST  DEGREE.  83 

r' 
2d.  What  remains  after  giving  to  the  second  child     2a-] 

/            (n—iy 
can  be  represented  by     r  — — r  or . 

Moreover,  this  remainder  also  forms  the  third  part.     Therefore 

we  have 

(n— l)r'  San 

-=Sa;  whence  /=■ 


n  n— 1 

3an  2a 

Then  the  second  part  is  2a-\ --^n=2a-j- -,    or  convert- 

^  n— 1  «— 1 

2«n+of 
ing  the  whole  number  into  a  fraction,  and  reducing,  — . 

r 
3d.  The  remainder,  after  giving  to  the  first  a-] ,    can  be  ex- 

r            (n — l)r 
pressed  by  r or  .  Now  this  remainder  should  form 

2an-\-a 
the  two  other  parts,  or  3a-) — . 

_                      (n—l)r  2an-\-a      ban— 2a 

Therefore,     -^ '—=2a+-  - 


11  n  —  \  11—1 

ban — 2a           n           5an^ — 2aii 
Hence,  r= —  X  7 ^77-=  — -, r-^ — . 

And  consequently  the  first  part  is 

5an^—2an  ban— 2a 

«H 7 T^ — -^n=a-\- 


{n-lf      ■  "~"^  (n-iy  ' 
ban — 2a  an" -{-San — a 

~""^n2-2n  +  l   "^     n"—2n+l    ' 

Ther  the  whole  estate  is 

2an-\-a       an" -{-San— a 
Sa+ V+- 


n— 1  n^— 2n+l 

Or,  by  reducing  the  whole  number  and  fractions  to  a  common 
denominator, 


84  ALGEBRA. 

3a(w^— 2w+l)  +  (2an+a)  (n  —  l)+a7v'  +  San—a 

H^— 27J  +  1  * 

Or  performing  the  operations  indicated  and  reducing 
6an^  —  4an+a  _o(6?j='— 4n+l) 
7i^-2rt+l     ~        (n-lf        ' 
which  agrees  with  the  preceding  result. 

This  solution  is  more  complete  than  the  preceding,  since  we  obtain 
from  it  the  estate  of  the  father,  and  the  expressions  for  the  three 
parts. 

9.  A  father  ordered  in  his  will,  that  the  eldest  of  his  children 
should  have  a  sum  a,  out  of  his  estate,  plus  the  «th  part  of  the  re- 
mainder ;  that  the  second  should  have  a  sum  2a,  plus  the  nth  part  of 
what  remained  after  having  subtracted  from  it  the  first  part  and  2a ; 
that  the  third  should  have  a  sum  Sa,  plus  the  nth  part  of  the  new  re- 
mainder — and  so  on.  It  is  moreover  supposed  that  the  children 
share  equally.  Required,  the  value  of  the  father's  estate,  the 
share  of  each  child,  and  the  number  of  children. 

This  problem  is  remarkable,  because  the  number  of  conditions  con- 
tained  in  the  enunciation  is  greater  than  the  number  of  unknown 
values  required  to  be  found. 

Let  the  estate  of  the  father  be  represented  by  x  :   then  will  x—a 

express  what  remains  after  having  taken  from  it  the  sum  a.    There- 

fo)-e  the  share  of  the  eldest  is 

x—a         an-\-x—a 

a-\- or  =lst.  part. 

n  n  ^ 

Subtracting  the  first  part,  and  2a,  from  a;,  we  have 

-x—a)  nx— 

^or, 

n 

nx—^an  —  x-\-a 
the  nth  part  of  which  is, .  tt 

Hence,  the  share  of  the  second  child  is 

nx  —  San  —  x+a           2an^-\-nx—^an  —  r+a 
2a-] ,  or  ■ ^ =  2d  part. 


(an-{-x—a)  nx—San—x-{-a 

2a — ^^ or, 

n  n 


EQUATIONS  OF  THE  FIRST  DEGREE.  85 

In  like  manner,  the  other  parts  might  be  formed,  but  as  all  the 
parts  should  be  equal,  it  suffices  to  form  the  equation  of  the  problem, 
to  equate  the  two  first  parts,  which  gives 

an + a; — a      2an^-\-  nx — Ban —x-\-a 
n        ~  r?  ' 

whence, 

x=an^—2an+a=a{n—  If. 

Substituting  this  value  of  a;  in  the  expression  for  the  first  part, 
we  find 

an + an^ — 2an  -{-a  — a 


or  reducing. 


71 


■an—a=^a{n  —  \) ; 


and  as  the  parts  are  equal,  by  dividing  the  whole  estate  by  the  first 

part,  we  will  obtain  a  quotient  that  will  show  the  number  of  child. 

an^—2an-\-a  ,  ,  ,         ^ 

ren;  therefore,    ,   or  n—l,  denotes  the  number  of 

an— a 

children. 

The  father's  estate,  .         .         an" —2an-\-a=a{n  —  lf. 

The  share  of  each  child,  .         rt(w— 1). 

Whole  number  of  children,        .  {i^—^)' 

It  yet  remains  to  be  shown,  that  the  other  conditions  of  the  pro. 
blem  are  satisfied  ;  that  is,  that  by  giving  to  the  second  child,  2a  plus 
the  nth  part  of  what  remains  ;  to  the  third,  2>a  plus  the  nth  part  of 
what  remains,  &c.,  the  share  of  each  child  is  in  fact  (n— 1)«. 

The  difference  between  the  estate  of  the  father  and  the  first  part 
being     «(n— 1)^— a(n— 1),     the  share  of  the  second  child  will  be 

a(n-lY-a(n-\)-2a  2a(n-l)+a{n-lf-a{n-\) 

2a-\-- ^ ,  or  -^ ^ ^ -, 

n  n 

and  reducing 

a(n-\)-\-a{n-\y  a{n-\)  {\+n-l) 

or  , 

n  n 

8 


or  fl(n  — 1). 

In  like  manner,  the  difference  between  a{n—iy  and  the  two  first 

parts  being,     a{n—iy—2a{n—l),     the  third  part  will  be 

a{n-lY-2a{n-l)-2a 

Sa-\ , 

n 

which  being  reduced,  becomes 

a{n-l)+a{n-iy 


a(n-l). 


In  the  same  way  we  would  obtain  for  the  fourth  part 

fl(7i_l)2_3rt(,i_l)_4a  a(,n—l)+ci(.7i—iy 

4a -1 ,  or ,    and  so  on. 

71  n 

Hence  all  the  conditions  of  the  enunciation  are  satisfied. 

10.  What  number  is  that  from  which,  if  5  be  subtracted,  |  of 
the  remainder  will  be  40  ?  Am.     65. 

11.  A  post  is  \  in  the  mud,  i  in  the  water,  and  ten  feet  above  the 
water  :  what  is  the  whole  length  of  the  post  ? 

Ans.     24  feet. 

12.  After  paying  I  and  \  of  my  money,  I  had  66  guineas  left  in 
my  purse :  how  many  guineas  were  in  it  at  first  ? 

Ans.     120. 

13.  A  person  was  desirous  of  giving  3  pence  a  piece  to  some 
beggars,  but  found  he  had  not  money  enough  in  his  pocket  by  8 
pence  :  he  therefore  gave  them  each  2  pence  and  had  3  pence  re- 
maining  :  required  the  number  of  beggars.  Ans.     11. 

14.  A  person  in  play  lost  \  of  his  money,  and  then  won  3  shil- 
lings ;  after  which  he  lost  \  of  what  he  then  had ;  and  this  done, 
found  that  he  had  but  12  shillings  remaining  :  what  had  he  at  first  ? 

Ans.     205. 

15.  Two  persons,  A  and  B,  lay  out  equal  sums  of  money  in  trade  ; 
A  gains  $126,  and  B  loses  $87,  and  A's  money  is  now  double  of  B's  : 
what  did  each  lay  out  ?  Ans.     $300. 

16.  A  person  goes  to  a  tavern  with  a  certain  sum  of  money  in  his 
pocket,  where  he  spends  2  shillings  ;  he  then  borrows  as  much  mo- 


EQUATIONS  OF  THE  FIRST  DEGREE.  87 

ney  as  he  had  left,  and  going  to  another  tavern,  he  there  spends  2 
shillings  also  ;  then  borrowing  again  as  much  money  as  was  left,  he 
went  to  a  third  tavern,  where  likewise  he  spent  two  shillings  and 
borrowed  as  much  as  he  had  left ;  and  again  spending  2  shillings 
at  a  fourth  tavern,  he  then  had  nothing  remaining.  What  had  he 
at  first  ?  -471S.     3*.  9d. 

Of  Equations  of  the  First  Degree  involving  two  or  more 
unknown  quantities. 

95.  Although  several  of  the  questions  hitherto  resolved,  contain- 
ed in  their  enunciation  more  than  one  unknown  quantity,  we  have 
resolved  them  by  employing  but  one  symbol.  The  reason  of  this 
is,  that  we  have  been  able,  from  the  conditions  of  the  enunciation, 
to  express  easily  the  other  unknown  quantities  by  means  of  this  syra- 
bo] ;  but  this  is  not  the  case  in  all  problems  containing  more  than 
one  unknown  quantity. 

To  ascertam  how  problems  of  this  kind  are  resolved :  first,  take 
some  of  those  which  have  been  resolved  by  means  of  one  unknown 
quantity. 

1.  Given  the  sum  a,  of  two  numbers,  and  their  difference  h,  it  is 
required  to  find  these  numbers. 

Let  X—  the  greater,  and  y  the  less  number.  | 

Then  by  the  conditions    ....         x+y=  a. 
and  ....         x—y=h. 

By  adding  (Art.  86.  Ax.  1.)    .         .         .  2x=a+i. 

By  subtracting  (Art.  86.  Ax.  2.)     .         .  2y—a—b. 

Each  of  these  equations  contains  but  one  unknown  quantity. 

a+b 

From  the  first  we  obtain  .         .         •  x=         . 

a  —  b 
And  from  the  second      ....  u=— -— . 


88  ALGEBRA. 

Verification, 
a+i     a  —  b     2a  ,  o  +  i      a  —  b     2b 

-2-+^-=¥  =  '^'  ^"^  "2 ^=Y=^- 

For  a  second  example,  let  us  also  take  a  problem  that  has  been 
already  solved. 

2.  A  person  engaged  a  workman  for  48  days.  For  each  day 
that  he  labored  he  was  to  receive  24  cents,  and  for  each  day  that  he 
was  idle  he  was  to  pay  12  cents  for  his  board.  At  the  end  of  the 
48  days,  the  account  was  settled,  when  the  laborer  received  504 
cents.  Required  the  number  of  workmg  days  and  the  number  of 
days  he  was  idle. 

Let  X  —  the  number  of  working  days. 

y  =  the  number  of  idle  days. 
71  —  the  whole  number  of  days  =  48. 
a  =  what  he  received  per  day  for  work  =  24  cts. 
b  =   what  he  paid  per  day  for  board  =  12  cts. 
c  =   what  he  received  at  the  end  of  the  time  =  504. 
Then,     ax  =  what  he  earned, 
And        by  =  what  he  paid  for  his  board. 

x-\-  y=n. 


We  have  by  the  question      .         .         .  .  , 

•'  ^  (  ax—by:=  c. 

It  has  already  been  shown  that  the  two  members  of  an  equation 
can  be  multiplied  by  the  same  number,  without  destroying  the  equal- 
ity  ;  therefore  the  two  members  of  the  first  equation  may  be  multi- 
plied  by  b,  the  co-efRcient  of  y  in  the  second,  and  we  have 

The  equation     .....         bx-{-by=bji. 

Which,  added  to  the  second  .         .         ax—  by=  c. 

Gives         ......         ax-\-bx^bn-\-c. 

bn-\-c 
Whence   ......  a;= — — ;. 

a+b 

In  like  manner,  multiplying  the  two  members  of  the  first  equa- 
tion by  a,  tlie  co-efficient  of  x  in  the  second,  it  becomes 


EQUATIONS  OF  THE  FIRST  DEGREEo  89 


From  which,  subtract  the  second  equation, 
And  we  obtain 

Whence 

By  introducing  a  symbol  to  represent  each  of  the  unknown  quan- 
titles  in  the  preceding  problem,  the  solution  which  has  just  been 
given  has  the  advantage  of  making  known  the  two  required  num. 
bers,  independently  of  each  other. 

Elimination. 

96.  The  method  which  has  just  been  explained  of  combining  two 
equations,  involving  two  unknown  quantities,  and  deducing  there- 
from a  single, equation  involving  but  one,  may  be  extended  to  three, 
four,  or  any  number  of  equations,  and  is  called  elimination. 
There  are  three  principal  methods  of  elimination : 
1st.     By  addition  and  subtraction. 
2d.      By  substitution. 
3d.      By  comparison. 
We  will  consider  these  methods  separately. 

Eliminalion  hy  Addition  and  Subtraction. 


97.  Take  the  two  equations  .         .  ,  ,, 

which  may  be  regarded  as  the  algebraic  enunciation  of  a  problem 
containing  two  unknown  quantities.  If,  in  these  equations,  one  of 
the  unknown  quantities  was  affected  with  the  same  co-efRcient,  we 
might,  by  a  simple  subtraction,  form  a  new  equation  which  would 
contain  but  one  unknown  quantity,  and  from  which  the  value  of  this 
unknown  quantity  could  be  deduced. 

Now,  if  both  members  of  the  first  equation  be  multiplied  by  9, 
the  co-efficient  of  y  m  the  second,  and  the  two  members  of  the 
second  by  7,  the  co-efficient  of  y  in  the  first,  we  will  obtain 
8* 


90  ALGEBRA. 

45a,'+63^=:387, 

equations  which  may  be  substituted  for  the  two  first,  and  in  which 
y  is  affected  with  the  same  co-efficient. 

Subti'acting,  then,  the  first  of  these  equations  from  the  second, 
there  results  32a;=96,  whence  x=3. 

Again,  if  we  muhiply  both  members  of  the  first  equation  by  11, 

the  co-efficient  of  x  m  the  second,  and  both  members  of  the  second 

by  5,  the  co-efficient  of  x  in  the  first,  we  will  form  the  two  equations 

55a;+77?/=473, 


o/ic;   \      which  may  be  substituted  for  the  two 

ODX-\-HtOy — o45,   ; 

proposed  equations,  and  in  which  the  co-efficients  of  «  are  the  same. 

Subtracting,  then,  the  second  of  these  two  equations  from  the  first, 
there  results       32^=128,  whence  2/=4. 

Therefore   a;=3  and  y=,4:,  are  the  values  of  x  and  y,  which 
should  verify  the  enunciation  of  the  question.     Indeed  we  have, 
1st.       5x3+7x4=15+28=43; 
2d.      11x3  +  9x4=33+36=69. 

The  method  of  elimination,  just  explained  is  called  the  method  by 
addition  and  subtraction,  because  the  unknown  quantities  disappear 
by  additions  and  subtractions,  after  having  prepared  the  equations 
in  such  a  manner  that  one  unknown  quantity  shall  have  the  same 
co-efficient  in  two  of  them. 

Elimination  by  Substitution. 

5a;+7t/=43. 


98.  Take  the  same  equations         .         .  #  , ,     .  „ 

^  (  lla;+9j/=69. 

Find  the  value  of  x  in  the  first  equation,  which  gives 

43-7y 

Substitute  this  value  of  x  m  the  second  equation,  and  we  have 

43 -7y 
nX—~+9y=Q9. 


EQUATIONS  OF  THE  FIRST  DEGREE. 


91 


or 

or 

Hence 

And 


473-77?/+45j/=345. 

—  d2y  =  —  l28. 

y=4. 

43-28 


This  method,  called  the  method  by  substitution,  consists  in  finding 
the  value  of  one  of  the  unknown  quantities  m  one  of  the  equations, 
as  if  the  other  unknown  quantities  were  already  determined,  and  in 
substituting  this  value  in  the  other  equations;  in  this  way  new  equa- 
tions  are  formed,  which  contain  one  unknown  quantity  less  than  the 
given  equations,  and  upon  which  we  operate  as  upon  the  proposed 
equations. 


Elimination  by  Comparison. 

5x+72/=43 
llx+9y=e9. 
Finding  the  value  of  a;  in  the  first  equation,  we  have 
43 -7z/ 


99.  Take  the  same  equations 


have 


And  finding  the  value  of  a;  in  the  second,  we  obtain 
69  — 9y 

Let  these  two  values  of  x  be  placed  equal  to  each  other,  and  we 
43—   7y_   69-   9y 

~      IT 


Or,  . 
Or,  . 
Hence, 

And, 


473-771/ =  345-45?/ 
-d2y  =  -128. 
y=  4 
69-36 


11 


This  method  of  elimination  is  called  the  method  by  comparison, 
and  consists  in  finding  the  value  of  the  same  unknown  quantity  in  all 
the  equations,  placing  them  equal  to  each  other,  two  and  two,  which 


92  ALGEBRA. 

necessarily  gives  a  new  set  of  equations,  containing  one  unknown 
quantity  less  than  the  other,  upon  which  we  operate  as  upon  the 
proposed  equations. 

But  there  is  an  inconvenience  in  the  two  last  methods,  which  the 
method  by  addition  and  subtraction  is  not  subject  to,  viz. :  they  pro- 
duce  new  equations,  containing  denominators,  which  it  is  afterwards 
necessary  to  make  disappear.  The  metliod  by  substitution  is,  how- 
ever,  advantageously  employed  whenever  the  co-efficient  of  one  of 
the  unknown  quantities  is  equal  to  unity  in  one  of  the  equations,  be- 
cause then  the  inconvenience  of  which  we  have  just  spoken  does  not 
occur.  We  shall  sometimes  have  occasion  to  employ  it,  but  gene- 
rally, the  method  by  addition  and  subtraction  is  preferable.  It  more- 
over  presents  this  advantage,  viz.  :  when  the  co-efficients  are  not 
too  great,  we  can  perform  the  addition  or  subtraction  at  the  same 
time  with  the  multiplication  which  is  necessary  to  render  the  co-ef- 
ficients  equal  to  each  other. 

100.  Let  us  now  consider  the  case  of  three  equations  involving 
three  unknown  quantities. 

{5a;-6j/+42=15. 
7x-{-4^y—Sz=19. 
2x+  ?/+6s=46. 
To  eliminate  2  b)'  means  of  the  first  two  equations,  multiply  the 
first  by  3  and  the  second  by  4,  then  since  the  co-efficients  of  z  have 
contrary  signs,  add  the  two  results  together :    this  gives   a   new 
equation  ......  43a;— 2]/=  121  " 

Multiplying  the  second  equation  by  2,  a  fac- 
tor of  the  co-efficient  of  s;  in  the  third  equation, 
and  adding  them  together,  we  have    .         .  16x-\-9y=   84 

The  question  is  then  reduced  to  finding  the  values  of  x  and  y, 
which  will  satisfy  these  new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and  the  re- 
sults be  added  together,  we  find 

419a;=rl257,  whence  x=3. 


EQUATIONS  OF  THE  FIRST  DEGREE.  93 

We  might,  by  means  of  the  two  equations  involving  x  and  y,  de- 

termiiie  y  in  the  same  way  we  have  determined  x  ;  but  the  value  of 

y  may  be  determined  more  simply,  by  observing  that  the  last  of 

these  two  equations  becomes,  by  substituting  for  x  its  value  found 

above, 

84-48 
48+9?/=84     whence     y= — — =4. 

In  the  same  manner  the  first  of  the  three  proposed  equations,  be- 

comes,  by  substituting  the  values  of  x  and  y, 

24 
15—24+4^=15,     whence     z—~=6. 

101.  Hence,  if  there  are  m  equations  involving  a  like  number  of 
unknown  quantities,  the  unknown  quantities  may  be  eliminated  by 
the  following 

RULE. 

I.  To  eliminate  one  of  the  unknown  quantities,  comhine  any  one  of 
the  equations  with  each  of  the  m— 1  others ;  there  will  thu^  he  obtain- 
ed m— 1   new  equations  containing  m— 1  unknown  quantities. 

II.  Eliminate  another  unknown  quantity  hy  combining  one  of  these 
new  equations  with  the  m— 2  others ;  this  will  give  m— 2  equations 
containing  m— 2  unknoion  quantities. 

III.  Continue  this  series  of  operations  until  a  single  equation  con. 
taining  but  one  unknown  quantify  is  obtained,  from  which  the  value  of 
this  unknoion  quantity  is  easily  found.  Then  hy  going  back  through 
the  series  of  equations  which  have  been  obtained,  the  values  of  the 
other  unknoion  quantities  may  be  successively  determined. 

102.  It  often  happens  that  each  of  the  proposed  equatiotis  does 
not  contain  all  the  unknown  quantities.  In  this  case,  with  a  little 
address,  the  elimination  is  very  quickly  performed. 

Take  the  four  equations  involving  four  unknown  quantities  : 
2x-Zy+2z=\Z\       .     .     (1)         4i/+2sr=14     .     .     (3). 
^u-2x=Zo\       .     .     (2)         5i/  +  3M=32     .     .     (4). 


By  inspecting  these  equations,  we  see  that  the  ehmination  of  %  in 
the  two  equations,  (1)  and  (3),  will  give  an  equation  involving  .r  and 
y ;  and  if  we  eliminate  u  in  the  equations  (2)  and  (4),  we  will  ob- 
tain a  second  equation,  involving  x  and  ?/.  These  two  last  unknown 
quantities  may  therefore  be  easily  determined.  In  the  first  place, 
the  elimination  of  z  in  (1)  and  (3)  gives       .         .     7^— 2x=l 

That  of  M  in  (2)  and  (4),  gives        .  .  .  20j/+6a;=38 

Multiplying  the  first  of  these  equations  by  3, 
and  adding 

Whence 

Substituting  this  value  in  7?/— 2x=l,  we  find 

Substituting  for  x  its  value  in  equation  (2), 
it  becomes  4m— 6=30,  whence 

And  substituting  for  y  its  value  in  equation 
(3),  there  results      ...... 


41t/=41 

y=  1 

x=   3 


EXAMPLES.                                                                       yy 

1.  Given 

2a;+3?/=16,  and  2>x—2y=\\     to  find  the  values  of 

a;  and  y. 

Ans.     a;=5,  3/=2. 

2.  Given 

'2x     3i/       9            3x     2j/        61 
5  +  4  ~20  ^"^    4  +  5  =  120      ^°  ^^  ^^'^  ''^^"'' 

of  X  and  y. 

1             1 

Ans.     x^—,  y=Y- 

3.  Given 

X                                 y 
y+7?/=99,  and  Y+7a;=51,     to  find  the  values  of 

X  and  y. 

Ans.     xz=zl,  y=\A. 

4.  Given 

^-..=-^-,8,  .a  t''+:-B=v+-. 

to  find  the  values  of  x  and  y.                     Ans,     x—%^,   ?/=40. 

'      ^+     y+     z=29^ 

5.    Given 

x-\-  2*/+   32=62  1       ,    ^    ,               , 
<                                       >      to  find  X,  y  and  z. 

Am.     x=Q,  y=9,  z=12. 


EQUATIONS  OF  THE  FIRST  DEGREE. 


95 


6.    Gi 


7.    Given      <' 


2x+  Ay—  3z=22 
4a;—  2y-\-  5z  =  18 
6a;+   7y—     2=63 


"+Y^+T^=3"^ 


T^'+T^+-5-^=^''^ 


to  find  a.',  y  and  t. 
iln5.     a;=3,  y='!,  2=4. 

>      to  find  X,  y  and  z. 


T^+T^+^ 


12 
2z+   3w=17^ 


x=12,  y=20,  5=30. 


>      to  find  X,  y,  z,  u,  and  I. 


Ix- 

Ay—  2z+  <=11 
8.  Given  <f  .5^—  3.c-  2u—  8 
4?/—  3if+  2/  =  9 
3z+  8u=33^ 
^ns.  a;=2,  ]/=4,  z=3,  u=3,  t=l. 
103.  In  all  the  preceding  reasoning,  we  liave  supposed  the  num- 
ber of  equations  equal  to  the  number  of  symbols  employed  to  de- 
note the  unknown  quantities.  This  must  be  the  case  in  every  pro- 
blem involving  two  or  more  unknown  quantities,  in  order  that  it  may 
be  determinate ;  that  is,  in  order  that  it  may  not  admit  of  an  infi- 
nite  number  of  solutions. 

Suppose,  for  example,  that  a  problem  involving  two  unknown 
quantities,  a;  and  y,  leads  to  the  single  equation,  5a;— 3^=12  ;  we 


deduce  from  it 

a;= r — .     Now,  by 

o 

mak 

2/=l,      2,      3,      4, 

5, 

there  results, 

18     21     24 

27 

^-^'    y    -5-'    T' 

"5" 

-ITi     -r-j    -r-,     6,    (Sjc 


and  every  system  of  values. 


96  ALGEBRA. 

18\  21\ 

(y=l,x=S);     (y=2,x=-);     (y=3,  x=-)  ;  6^c. 

substituted  for  x  and  y  in  the  equation,  will  satisfy  it  equally  well. 

If  we  had  two  equations  involvuig  three  unknown  quantities,  we 
could  in  the  first  place  eliminate  one  of  the  unknown  quantities  by 
means  of  the  proposed  equations,  and  thus  obtain  an  equation,  which, 
contaming  two  unkno\vTi  quantities,  would  be  satisfied  by  an  infinite 
number  of  systems  of  values  taken  for  these  unkno^vn  quantities. 
Therefore,  in  order  that  aproUem  may  he  determined,  its  enunciation 
must  contain  at  least  as  many  different  conditions  as  there  are  unknown 
quantities,  and  these  conditions  must  he  such,  thai  each  of  them  may 
be  expressed  by  an  independent  equation ;  that  is,  an  equation  not 
produced  by  any  combination  of  the  others  of  the  system. 

If,  on  the  contrary,  the  number  of  independent  equations  exceeds 
the  number  of  unknown  quantities  involved  in  them,  the  conditions 
which  they  express  cannot  be  fulfilled. 

For  example,  let  it  be  required  to  fijid  two  numbers  such  that 
their  sum  shall  be  100,  their  difference  80,  and  their  product  700. 

The  equations  expressing  these  conditions  are, 
x+y=\QO 
x—y=  80 
and  a;x«/='700. 

Now,  the  first  two  equations  determine  the  values  of  x  and  y,  viz. 
x=90  and  3/=10.  The  product  of  the  two  numbers  is  therefore 
known,  and  equal  to  900.  Hence  the  third  condition  cannot  be  ful- 
filled. 

Had  the  product  been  placed  equal  to  900,  all  the  conditions 
would  have  been  satisfied,  in  which  case,  however,  the  third  would 
not  have  been  an  independent  equation,  since  the  condition  expressed 
by  it,  is  unplied  in  the  other  two. 

QUESTIONS. 

1.  What  fraction  is  that,  to  the  numerator  of  which,  if  1  be  add. 


EQUATIONS  OF  THE  FIRST  DEGREE.  97 

1 


ed,  its  value  will  be   — ,  but  if  one  be  added  to  its  denominator,  its 


1 

value  will  be  -;-.  ^    -,„+*-;«* 

Let  the  fraction  be  represented  by     — .  , 

^                                      a;+l       1         .      X         1 
Then,  by  the  question     =y  ^^^  ^TZY^T* 

Whence     Sx-{-S=y,  and  4:X=y-\-l. 

Therefore,  by  subtracting,     a;— 3  =  1  or  x=  4. 

Hence,       12+3=?/:         therefore  y=l5. 

2.  A  market  woman  bought  a  certain  number  of  eggs  at  2  for  a 

penny,  and  as  many  others,  at  3  for  a  penny,  and  having  sold  them 

again  altogether,  at  the  rate  of  5  for  2d,  found  that  she  had  lost 

4(Z  :  how  many  eggs  had  she  ? 

=     the  whole  number  of  eggs. 

=     the  number  of  eggs  of  each  sort. 

=     the  cost  of  the  first  sort. 

=     the  cost  of  the  second  sort. 

4« 
2x  :  —     the  amount  for  which  the  eggs  were 
o 


1         1        4x 
Hence,by  the  question     — >r+— a; — —  =4. 

Therefore     .         .         15a;+10.T;-24.i-=120. 

Or,  .         .  x=  120     the  number  of  eggs  of 

each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars  for  which  he 
drew  a  certain  interest ;  but  he  owed  the  sum  of  20,000  dollars,  for 
which  he  paid  a  certain  interest.     The  intei-est  that  he  received  ex- 
ceeded that  which  he  paid  by  800  dollars.     Another  person  pos. 
9 


Let 

2x-- 

Then 

X-. 

Then  will 

1 

And 

1 

3^^ 

But     5  : 

2  :  : 

sold. 

98  ALGEBRA. 

sessed  35,000  dollars,  for  which  he  received  interest  at  the  second 
of  the  above  rates,  but  he  owed  24,000  dollars,  for  which  he  paid 
interest  at  the  first  of  the  above  rates.  The  interest  that  he  re- 
ceived  exceeded  that  which  he  paid  by  310  dollars.  Required,  the 
two  rates  of  interest. 

Let  X  and  y  denote  the  two  rates  of  interest :  that  is,  the  interest 
of  Si 00  for  the  given  time. 

To  obtain  the  interest  of  $30,000  at  the  first  rate  denoted  by  x, 

we  form  the  proportion 

30,000a; 
100   :  a;  :  :  30,000  :  :   —vw^    or  300a;. 

And  for  the  interest  $20,000,  the  rate  being  y. 

20,000y 
100  :  3/  :  :  20,000  :  :   -JT^  or  200y. 

But  from  the  enunciation,  the  difference  between  these  two  in- 
terests  is  equal  to  800  dollars. 

We  have,  then,  for  the  first  equation  of  the  problem, 
300x-200(/=800. 

By  writing  algebraically  the  second  condition  of  the  problem,  we 
obtain  the  other  equation, 

350?/-240x=310. 

Both  members  of  the  first  equation  being  divisible  by  100,  and 
those  of  the  second  by  10,  we  may  put  the  following,  in  place  of 
them  : 

Sx—2y=8,         S5y—24x=3l. 

To  eliminate  x,  multiply  the  first  equation  by  8,  and  then  add  it 
to  the  second  ;  there  results 

l9y—95,  whence  i/=5. 

Substituting  for  y,  in  the  first  equation,  its  value,  this  equation 
becomes 

3a;— 10=8,  whence  a;=6. 

Therefore,  the  first  rate  is  6  per  cent.,  and  the  second  5. 


EQUATIONS  OF  THE  FIRST  DEGREE.  99 

Verification. 

$30,000,  placed  at  6  per  cent.,  gives     300x6,   =  81800. 
$20,000,      do.        5  do.  200x5,   =   81000. 

And  we  have  1800  —  1000=800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  There  are  three  ingots  composed  of  different  metals  mixed 
together.  A  pound  of  the  first  contains  7  ounces  of  silver,  3  ounces 
of  copper,  and  6  of  pewter.  A  pound  of  the  second  contains  12 
ounces  of  silver,  3  ounces  of  copper,  and  1  of  pewter.  A  pound 
of  the  third  contains  4  ounces  of  silver,  7  ounces  of  copper,  and  5 
of  pewter.  It  is  required  to  find  how  much  it  will  take  of  each  of 
the  three  ingots  to  form  a  fourth,  which  shall  contain  in  a  pound,  8 
ounces  of  silver,  3|  of  copper,  and  4|  of  pewter. 

Let  X,  y  and  z  represent  the  number  of  ounces  which  it  is  neces- 
sary to  take  from  the  three  ingots  respectively,  in  order  to  form  a 
pound  of  the  required  ingot.  Since  there  are  7  ounces  of  silver  in 
a  pound,  or  16  ounces,  of  the  first  ingot,  it  follows  that  one  ounce 
of  it  contains  -^^  of  an  ounce  of  silver,  and  consequently  in  a  num. 

7x 
ber  of  ounces  denoted  by  x,  there  is     —     ounces  of  silver.  In  the 

12^  Az 

same  manner  we  would  find  that     — — -  and  — -,  express  the  num. 

lb  16       '^ 

ber  of-  ounces  of  silver  taken  from  the  second  and  third,  to  form 

the  fourth  ;  but  from  the  enunciation,  one  pound  of  this  fourth  ingot 

contains  8  ounces  of  silver.     We  have,  then,  for  the  first  equation 

7a;      \2y      Ax 

16       16       16^ 

or,  making  the  denominators  disappear.     .     7x-f  12?/+4z=128 

As  respects  the  copper,  we  should  find    .     .     3a'+   3^+7?=  60 

and  with  reference  to  the  pewter      .     .     .     Qx-\-     ?/4-52=  68 

As  the  co-efficients  of  y  in  these  three  equations,  are  the  most 

simple,  it  is  most  convenient  to  eliminate  this  unknown  quantity  first. 


100  ALGEBRA. 

JMultiplyiiig  the  second  equation  by  4,  and  subtracting  the  first 
equation  from  the  product,  we  have      .     .     .       5ic+24z=112 

Multiplying  the  third  equation  by  3,  aiid 
subtracting  the  second  from  the  product     .     .     15a;+  82=144 

Multiplying  this  last  equation  by  3,  and  subtracting  the  preced- 
ing one  from  the  product,  we  obtain  40a;=320,  whence  x=8. 

Substitute  this  value  for  x  m  the  equation  15x+82=rl44  ;  it  be- 
comes 

120  +  82=144,     whence     z=3. 

Lastly,  the  two  values  x=8,  «=3,  being  substituted  in  the  equa- 
tion 6a;+2/+52=68,  give  48+?/+15=68,  whence  2/=5. 

Therefore  in  order  to  form  a  pound  of  the  fourth  ingot,  we  must 
take  8  ounces  of  the  first,  5  ounces  of  the  second,  and  3  of  the 
third. 

Verification. 
If  there  be  7  ounces  of  silver  in  16  ounces  of  the  first  ingot,  in 
8  ounces  of  it,  there  should  be  a  number  of  ounces  of  silver  ex- 

pressed  by     —r^' 

12x5  ,     4X3 

In  like  manner     — — —     and     — r^-     will  express  the  quantity 
lb  lb 

of  silver  contained  in  5  ounces  of  the  second  ingot,  and  3  ounces  of 

the  third. 

7x8      12x5      4x3      128      ^        ^       ^ 

Now,  we  have    -r^-i r^ 1 — r^=-rz-=8;    therefore,  a 

16  16  16  lb 

pound  of  the  fourth  ingot  contains  8  ounces  of  silver,  as  required  by 
the  enunciation.  The  same  conditions  may  be  verified  relative  to 
the  copper  and  pewter. 

5.  What  two  numbers  are  those,  whose  difference  is  7,  and  sum 
33?  Ans.      13  and  20. 

6.  To  divide  the  number  75  into  two  such  parts,  that  three  times 
the  greater  may  exceed  seven  times  the  less  by  15. 

Ans,     54  and  21. 


EQUATIONS  OF  THE  FIRST  DEGREE.  101 

7.  In  a  mixture  of  wine  and  cider,  i  of  the  whole  plus  25  gal- 
lons was  wine,  and  i  part  minus  5  gallons  was  cider ;  how  many 
gallons  were  there  of  each  ? 

Ans.     85  of  wine,  and  35  of  cider. 

8.  A  bill  of  £120  was  paid  in  guineas  and  moidores,  and  the 
number  of  pieces  of  both  sorts  that  were  used  was  just  100  ;  if  the 
guinea  be  estimated  at  21*.  and  the  moidore  at  27  s.  how  many 
were  there  of  each  ?  Ans.     50  of  each. 

►  9.  Two  travellers  set  out  at  the  same  time  from  London  and 
York,  whose  distance  apart  is  150  miles ;  one  of  them  goes  8  miles 
a^day,  and  the  other  7  ;  in  what  time  will  they  meet  ? 

Ans.     In  10  days. 

10.  At  a  certain  election,  375  persons  voted  for  two  candidates, 
and  the  candidate  chosen  had  a  majority  of  91 ;  how  many  voted 
for  each  ?  Ans.     283  for  one,  and  142  for  the  other. 

11.  A's  age  is  double  of  B's,  and  B's  is  triple  of  C's,  and  the  sum 
of  all  their  ages  is  140  ;  what  is  the  age  of  each  1 

Ans.     A's=84,  B's=42,  andC's=zl4. 

12.  A  person  bought  a  chaise,  horse,  and  harness,  for  £60 ;  the 
horse  came  to  twice  the  price  of  the  harness,  and  the  chaise  to  twice 
the  price  of  the  horse  and  harness ;  what  did  he  give  for  each  ? 

j'£13.    6s.  8d.     for  the  horse. 
Ans.      J  £  6.  135.  M.     for  the  harness. 
(^  £40.  for  the  chaise. 

13.  Two  persons,  A  and  B,  have  both  the  same  income  :  A  saves 
i  of  his  yearly,  but  B,  by  spending  £50  per  annum  more  than  A, 
at  the  end  of  4  years  finds  himself  £100  in  debt ;  what  is  their 
income?  Ans.     £125. 

14.  A  person  has  two  horses,  and  a  saddle  worth  £50  ;  nolv  if 
the  saddle  be  put  on-  the  back  of  the  first  horse,  it  will  make  his 
value  double  that  of  the  second  ;  but  if  it  be  put  on  the  back  of  the 
second,  it  will  make  his  value  triple  that  of  the  first ;  what  is  the 
value  of  each  horse  ? 

Ans.     One  £30,  and  the  other  £40. 


102        ^■■'  ALGEBRA. 

15.  To  divide  the  number  36  into  three  such  parts  that  \  of  the 
first,  i  of  the  second,  and  J  of  the  third,  may  be  all  equal  to  each 
otlier.  -'   '-  ■'  :  Ans.     8,  12,  and  16. 

16.  A  footman  agreed  to  serve  his  master  for  £8  a  year  and  a 
livery,  but  was  turned  away  at  the  end  of  7  months,  and  received 
only  £2.  135.  4d.  and  his  livery;  what  was  its  value?    ^ '^ 

"        "  Ans.     £4.  165. 

17.  To  divide  the  number  90  into  four  such  parts,  that  if  the  first 
be  increased  by  2,  the  second  diminished  by  2,  the  third  multiplied^ 
by  2,  and  the  fourth  divided  by  2,  the  sum,  difference,  product,  and 
quotient  so  obtained,  will  be  all  equal  to  each  other. 

Ans.     The  parts  are  18,  22,  10,  and  40. 

18.  The  hour  and  minute  hands  of  a  clock  are  exactly  together 
at  12  o'clock  ;  when  are  they  next  together? 

Ans.     111.  b-^jViin. 

19.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer  in  12 
days ;  but  when  the  man  was  from  home,  it  lasted  the  woman  30 
da3-s  ;  how  many  days  would  the  man  alone  be  in  drinking  it  ? 

Ans.     20  days. 

20.  If  A  and  B  together  can  perform  a  piece  of  work  in  8  days, 
A  and  C  together  in  9  days,  and  B  and  C  in  10  days :  how  many  days 
would  it  take  each  person  to  perform  the  same  work  alone  ? 

Ans.     A  14f|-  days,  B  17|f,  and  C  23/,-. 

21.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a  time 
expressed  by  J  ;  a  second  laborei',  the  work  c  in  a  time  d  ;  a  third, 
the  work  e,  in  a  time/.  It  is  required  to  find  the  time  it  would  take 
the  three  laborers,  working  together,  to  perform  the  work  g. 

•  An5.     <^- ^^jj^j^^fj^i^^' 

Application. 

a=21 ;  J=4  ]  c=35  ;  d=6  |  e=40  ;  /=12  ]  ^=191 ; 
r,^'  X  will  be  found  equal  to  12. 

22^  If  32  pounds  of  sea  water  contain  1    pound  of  salt,  how 


no 


EQUATIONS  OF  THE  FIRST  DEGREE.  103 

much  fresh  water  must  be  added  to  these  32  pounds,  in  order  that 
the  quantity  of  salt  contained  in  32  pounds  of  the  new  mixtui-e 
shall  be  reduced  to  2  ounces,  or  |  of  a  pound  ? 

Ans.     22A  lb. 

23.  A  number  is  expressed  by  three  figures ;  the  sum  of  these 
figures  is  11 ;  the  figure  in  the  place  of  units  is  double  that  in  the 
place  of  hundreds  ;  and  when  297  is  added  to  this  number,  the  sum 
obtamed  is  expressed  by  the  figures  of  this  number  reversed.  What 
is  the  number  ?  Ans.     326 

24.  A  person  who  possessed  100,000  dollars,  placed  the  greater 
part  of  it  out  at  5  per  cent,  interest,  and  the  other  part  at  4  per 
cent.  The  interest  which  he  received  for  the  whole  amounted  to 
4640  dollars.     Required,  the  two  parts. 

^     .^4:^/;,^^^^^'^    ^^      .  ,  ^^^^     64,000  and  36,000. 

25.  A  person  possessed  a  certain  capital,  which  he  placed  out  at 
a  certain  interest.  Another  person  who  possessed  10,000  dollars 
more  than  the  first,  and  who  put  out  his  capital  1  per  cent,  more 
advantageously  than  he  did,  had  an  income  greater  by  800  dollars. 
A  third  person  who  possessed  15,000  dollars  more  than  the  first, 
and  who  put  out  his  capital  2  per  cent,  more  advantageously  than 
he  did,  had  an  income  greater  by  1500  dollars.  Required,  the  capi- 
tals  of  the  three  persons,  and  the  three  rates  of  interest. 

Sums  at  interest,         ^30,000,     $40,000,     $45,000. 
Rates  of  interest,  4  5  6       per  cent. 

26.  A  banker  has  two  kinds  of  money ;  it  takes  a  pieces  of  the 

first  to  make  a  crown,  and  b  of  the  second  to  make  the  same  sum. 

Some  one  offers  him  a  crown  for  c  pieces.  How  many  of  each  kind 

must  the  banker  give  him  ? 

,     ,  .    ,     a(c—h)        ,  ,  .    ,     i(a—c) 

Ans.     Istkmd,    -^ — r^ ;  2d  kind,   -^ =-^. 

a—b  a—b 

27.  Find  what  each  of  three  persons  A,  B,  C,  are  worth,  know- 
ing; 1st,  that  what  A  is  worth  added  to  Z  times  what  B  and  C  are 
worth  is  equal  top ;  2d,  that  what  B  is  worth  added  to  m  times  what 


104  ALGEBRA. 

A  and  C  are  worth  is  equal  to  q ;  3d,  that  what  C  is  worth  added  to 
n  times  what  A  and  B  are  worth  is  equal  to  r. 

This  question  can  be  resolved  in  a  very  simple  manner,  by  intro- 
ducing  an  auxiliary  unknown  quantity  into  the  calculus.  This  un- 
known  quantity  is  equal  to  what  A,  B  and  C  are  worth. 

28.  Find  the  values  of  the  estates  of  six  persons,  A,  B,  C,  D,  E, 
F,  from  the  following  conditions  :  1st.  The  sum  of  the  estates  of  A 
and  B  is  equal  to  a  ;  that  of  C  and  D  is  equal  to  I ;  and  that  of  E  and 
F  is  equal  to  c.  2d.  The  estate  of  A  is  worth  m  times  that  of  C  ; 
the  estate  of  D  is  worth  n  times  that  of  E,  and  the  estate  of  F  is 
worth  p  times  that  of  B. 

This  problem  may  be  resolved  by  means  of  a  single  equation, 
involving  but  one  unknown  quantity. 

Theory  of  Negative  Quantities.     Explanation  of  the  terms, 
Nothing  and  Infinity. 

104.  The  algebraic  signs  are  an  abbreviated  language.  They 
point  out  in  the  shortest  and  clearest  manner  the  operations  to  be 
performed  on  the  quantities  with  which  they  are  connected. 

Having  once  fixed  the  particular  operation  indicated  by  a  parti- 
cular  sign,  it  is  obvious  that  that  operation  should  always  be  perform- 
ed on  every  quantity  before  which  the  sign  is  placed.  Indeed,  the 
principles  of  algebra  are  all  established  upon  the  supposition,  that 
each  particular  sign  which  is  employed  always  means  the  same 
thing ;  and  that  whatever  it  requires  is  strictly  performed.  Thus, 
if  the  sign  of  a  quantity  is  +,  we  understand  that  the  quantity  is  to 
be  added  ;  if  it  is  —,  we  understand  that  it  is  to  be  subtracted. 

For  example,  if  we  have  —4,  we  understand  that  this  4  is  to  be 
subtracted  from  some  other  number,  or  that  it  is  the  result  of  a  sub- 
traction in  which  the  number  to  be  subtracted  was  the  greatest. 

If  it  were  required  to  subtract  20  from  16,  the  subtraction  could 
not  be  made  by  the  rules  of  arithmetic,  since  16  does  not  contain 


EQUATIONS  OF  THE  FIRST  DEGREE.  105 

20  ;  nor  indeed  can  it  be  entirely  performed  by  Algebra.     We 
write  the  numbers  for  subtraction  thus, 

16  — 20=16-16— 4= -4. 
By  decomposing  —20  into  —16  and  —4,  the  —16  will  cancel 
the  +16,  and  leave  —4  for  a  remainder. 

We  thus  indicate  that  the  quantity  to  be  subtracted  exceeds  the 
quantity  from  which  it  is  to  be  taken,  by  4. 

To  show  the  necessity  of  giving  to  this  remainder  its  proper  sign, 
let  us  suppose  that  the  difference  of  16  —  20  is  to  be  added  to  10. 
The  numbers  would  then  be  wi-itten 

16-20=-   4 
+  10         =  +  10 
26-20=+  6 
105.  If  the  sum  of  the  negative  quantities  in  the  first  member  of 
the  equation,  exceeds  the  sum  of  the  positive  quantities,  the  second 
member  of  the  equation  will  be  negative,  and  the  verification  of  the 
equation  will  show  it  to  be  so. 

For  example,  if  a—b=c, 

and  we  make  a=15  and  i=18,  c  will  be  =—3.  Now  the  essen- 
tial sign  of  c  is  different  from  its  algebraic  sign  in  the  equation. 
This  arises  from  the  circumstance,  that  the  equation  a—b=c  ex- 
presses  generally,  the  difference  between  a  and  h,  without  indicating 
which  of  them  is  the  greater.  When,  therefore,  we  attribute  par- 
ticular values  to  a  and  h,  the  sign  of  c,  as  well  as  its  value,  becomes 
known. 

We  will  illustrate  these  remarks  t^  a  few  examples. 
1.  To  find  a  number  which,  added  to  the  number  I,  gives  for  a 
sum  the  number  a. 

Let  a;=  the  required  number. 
Then,  by  the  condition  a?+J=a,  whence  x=a  —  b. 
This  expression,  or  formula,  will  give  the  algebraic  value  of  x  in 
all  the  particular  cases  of  this  problem. 

For  example,  let  rt=47,  i=29,  then  x=  47— 29=18. 


106  ALGEBRA. 

-     Agaiii,  let  fl=:24,  l=Sl ;  then  will  x=24  — 31  =  — 7. 

This  value  obtained  for  x,  is  called  a  negative  solution.  How  is 
it  to  be  interpreted  ? 

Considered  arithmetically,  the  problem  with  these  values  of  a  and 
b,  is  impossible,  since  the  number  b  is  already  greater  than  24.  Con- 
sidered  algebraically,  however,  it  is  not  so  ;  for  we  have  found  the 
value  of  a;  to  be  —7,  and  this  number  added,  in  the  algebraic  sense, 
to  31,  gives  24  for  the  algebraic  sum,  and  therefore  satisfies  both 
the  equation  and  enunciation. 

2.  A  father  has  lived  a  number  a  of  years,  his  son  a  number  of 
years  expressed  by  h.  Find  in  how  many  years  the  age  of  the 
son  will  be  one  fourth  the  age  of  the  father. 

Let  x=i     the  required  number  of  years. 

Then     a-\-x=     the  age  of  the  father 


^   ,  .  at  the  end  of  the  requir- 

and        b-\-x=     the  age  of  the  son      ' 

ed  time. 

a-\-x  a—^b 

Hence,  by  the  question    — — =Z»+a^ ;  whence  x= — - — . 

54-36     18 
Suppose  a=54,  and  b—9  :  then  x= — — — =——G. 

The  father  having  lived  54  years,  and  the  son  9,  in  6  years  the 
father  will  have  lived  60  years,  and  his  son  15  ;  now  15  is  the 
fourth  of  60  ;  hence,  a;=6  satisfies  the  enunciation. 

45-60 

Let  us  now  suppose  a=45,  and  h=zl5  :  then  x=^ — - — =-5. 

If  we  substitute  this  value  of  x  in  the  equation  of  condition,  we 
obtain 

45-5 
=15-5 

4 

or  10=10. 

Hence,  — 5  substituted  for  x  verifies  the  equation,  and  therefore 
is  the  true  answer. 

Now,  the  positive  result  which  has  been  obtained,  shows  that  the 


EQUATIONS  OF  THE  FIRST  DEGREE.  107 

age  of  the  father  will  be  four  times  that  of  the  son  at  the  expiration 
of  6  years  from  the  time  when  their  ages  were  considered  ;  while 
the  negative  result  indicates  that  the  age  of  the  father  was  four  times 
that  of  his  son,  5  years  previous  to  the  time  when  their  ages  were 
compared. 

The  question,  taken  in  its  most  general  or  algebraic  sense,  de- 
mands  the  time,  at  which  the  age  of  the  father  was  four  times  that 
of  the  son.  In  stating  it,  we  supposed  that  the  age  of  the  father 
was  to  be  augmented  ;  and  so  it  was,  by  the  first  supposition.  But 
the  conditions  imposed  by  the  second  supposition,  required  the  age 
of  the  father  to  be  diminished,  and  the  algebraic  result  confoi-med 
to  this  condition  by  appearing  with  a  negative  sign.  If  we  wished 
the  result,  under  the  second  supposition,  to  have  a  positive  sign,  we 
might  alter  the  enunciation  by  demanding  hoio  many  years  since  the 
age  of  the  father  ivasfour  times  that  of  his  son. 

If  x=  the  number  of  years,  we  shall  have 

a—x  4&  — a 

——  —  b—x:      hence     x= — r . 

If  a=45  and  5=15,     x  will  be  equal  to  5. 

Reasoning  from  analogy,  we  establish  the  following  general 
principles. 

1st.  Every  negative  value  found  for  the  unknown  quantity  in  a 
problem  of  the  first  degree,  will,  when  taken  vnth  its  proper  sign,  verify 
the  equation  from  which  it  was  derived. 

2d.  That  this  negative  value,  taken  with  its  proper  sign,  will  also 
satisfy  the  enunciation  of  the  problem,  understood  in  its  algebraic 
serise. 

3d.  If  the  emmciation  is  to  be  understood  in  its  arithmetical  sense, 
in  which  the  quantities  referred  to  are  always  supposed  to  be  positive, 
then  this  value,  considered  without  reference  to  its  sign,  may  be  con. 
spidered  as  the  answer  to  a  problem,  of  which  ike  enunciation  only  dif. 
fersfrom  that  of  the  proposed  problem  in  this,  that  certain  quantities 
which  were  additive,  have  become  subtractive,  and  reciprocally. 


108  ALGEBRA. 

106.  Take  for  example  the  problem  of  the  labourer  (Page.  88). 
Supposing  that  the  labourer  receives  a  sum  c,  we   have   the 
equations.  ''  ^-'' 

x+  y=n  )                            In^c  om—c 

whence    x- — r-,  3/=- 


ax—by=c  )  a-{-b  a+b 

But  if  we  suppose  that  the  labourer,  instead  of  receiving,  owes  a 
an  c,  the  equations  will  then  bo 

a;4-  y=n  ^  C  "x+  y=n, 


iy—ax=c  )        '     (  ax—by=  —  c. 
By  changing  the  signs  of  the  second  equation. 
Now  it  is  visible  that  we  can  obtain  immediately  the  values  of  x 
and  y,  which  correspond  to  the  preceding  values,  by  merely  chang- 
ing  the  sign  of  c  in  each  of  those  values  ;  this  gives 
bn—c  a7i-{-c 

'^~     a  +  b  '   ^~     a  +  f' 

To  prove  this  rigorously,  let  us  denote  —chyd; 

\    x-{-  y=n 
The  equations  then  become     j  _       and  they  only  differ 

from  those  of  the  first  enunciation  by  having  d  in  the  place  of  c. 
We  would,  therefore,  necessarily  find 

bn-\-d  an—d 

'^~  a+b   '  ^~   a-\-b    * 

And  by  substituting  —  c  for  d,  we  have 

bn-\-(  —  c)  an—(  —  c) 

""^     a+b      '  ^='   a+b       ' 

or  by  applying  the  rules  of  Art.  85, 

bn — c  an+c 

''-  a+b    '  ^^  a+b   ' 
The  results,  which  agree  to  both  enunciations,  may  be  compre- 
bended  in  the  same  formula,  by  writing 

bndzc  anrpc 


y=- 


a+b    '  ■"       a+b 


EQUATIONS  OF  THE  FIRST  DEGREE.  100 

The  double  sign  ±  is  read  phis  or  minus,  the  superior  signs  cor- 
respond to  the  case  in  which  the  labourer  received,  and  the  inferior 
signs  to  the  case  in  which  he  owed  a  sum  c.  These  formulas  com- 
prehend  the  case  in  which,  in  a  settlement  between  the  labourer  and 
his  employer,  their  accounts  balance.  This  supposes  c=0,  which 
gives 

in  an 

107.  When  a  problem  has  been  resolved  generally,  that  is,  by 
representing  the  given  quantities  by  letters,  it  may  be  required  to 
determine  what  the  values  of  the  unknown  quantities  become,  when 
particular  suppositions  are  made  upon  the  given  quantities.  The 
determination  of  these  values,  and  the  interpretation  of  the  peculiar 
results  obtained,  form  what  is  called  the  discussion  of  the  problem. 

The  discussion  of  the  following  question  presents  nearly  all  the 
circumstances  which  are  met  with  in  problems  of  the  first  degree. 

108.  Two  couriers  are  travelling  along  the  same  right  line  and 
in  the  same  direction  from  R'  towards  R.  The  number  of  miles 
travelled  by  one  of  them  per  hour  is  expressed  by  m,  and  the 
number  of  miles  travelled  by  the  other  per  hour,  is  expressed  by  n. 
Now,  at  a  given  time,  say  12  o'clock,  the  distance  between  them  is 
equal  to  a  number  of  miles  expressed  by  a  :  required  the  time  when 
they  will  be  together. 

R'  A B R. 

At  12  o'clock  suppose  the  forward  courier  to  be  at  B,  the  other 
at  A,  and  R  to  be  the  point  at  which  they  will  be  together. 
Then,    AB=rt,   their  distance  apart  at  12  o'clock. 
Let  .     .     t=      the  number  of  hours  which  must  elapse,  before 

they  come  together. 
And      .    a;=      the  distance  BR,  which  is  to  be  passed  over  by 

the  forward  courier. 
Then,  since  the  rate  per  hour,  multiplied  by  the  number  of  hours, 
will  give  the  distance  passed  over  by  each,  we  have, 
10 


110  ALGEBRA. 

iX»i  ■=  a+x=AR. 
iXn  —  X      =BR. 

Hence  by  subtractmg,  t{in—n)  —  a 

Thereforfe,  .         .         /  = . 

m—n 

Now  so  long  as  m>n,  i  will  be  positive,  and  the  problem  will  be 
solved  ill  the  arithmetical  sense  of  the  enunciation.  For,  if  niyn 
the"  courier  from  A  will  travel  faster  than  the  courier  from  B,  and 
will  therefore  be  continually  gaining  on  him :  the  interval  which 
separates  them  will  diminish  more  and  more,  until  it  becomes  0,  and 
then  the  couriers  will  be  found  upon  the  same  point  of  the  line. 

In  this  case,the  time  i,  which  elapses,  must  be  added  to  12  o'clock, 
to  obtain  the  time  when  they  are  together. 

But,  if  we  suppose  ni<n,  then  m—n  will  be  negative,  and  the 
value  of  t  will  be  negative.     How  is  this  result  to  be  interpreted  ? 

It  is  easily  explained  from  the  nature  of  the  question,  which,  con- 
sidered  in  its  most  general  sense,  demands  the  time  when  the 
couriers  are  together. 

Now,  under  the  second  supposition,  the  courier  which  is  in  ad- 
vance, travels  the  fastest,  and  therefore  will  continue  to  separate 
himself  from  the  other  courier.  At  12  o'clock  the  distance  between 
them  was  equal  to  a  :  after  12  o'clock  it  is  greater  than  a,  and  as 
the  rate  of  travel  has  not  been  changed,  it  follows  that  previous  to 
12  o'clock  the  distance  must  have  been  less  than  a.  At  a  certain 
hour,  therefore,  before  12  the  distance  between  them  must  have  been 
equal  to  nothing,  or  the  couriers  were  together  at  some  point  R'. 
The  precise  hour  is  found  by  subtracting  the  value  of  t  from  12 
o'clock. 

This  example,  therefore,  conforms  to  the  general  principle,  that, 
if  the  conditions  of  a  problem  are  such  as  to  render  the  unknown 
quantity  essentially  negative,  it  will  appear  in  the  result  uith  the 
minus  sign,  whenever  it  has  been  regarded  as  positive  in  the  enun- 
ciation. 


EQUATIONS  OF  THE  FIRST  DEGREE.  Ill 

If  we  wish  to  find  the  distances  AR,and  BR  passed  over  by  the 

two  couriers  before  coming  together,  we  may  take  the  equation 

a 

~m  —  n 

and  multiply  both  members  by  the  rates  of  travel  respectively  :  this 

will  give 

ma 
AR=m{= and 

711  —  71 

71  a 
BR=n/= . 

711  —  71 

ma 
Also,  .     .     AK'=—7}it= 

7ia 
and      .     .     BR' :^  —  nt= . 

771  — n 

from  which  we  see  that  the  two  distances  AR,  BR,  will  both  be 
positive  when  estimated  towards  the  right,  and  that  AR',  BR,'  will 
both  be  negative  when  estimated  in  the  contrary'  direction. 

109.  To  explain  the  terms  nothing  and  infinity,  let  us  consider 
the  equation 

a 

711 — ?i  ' 

If  in  this  equation  we  make  m=7i,  then  m— n=0,  and  the  value 
of  t  will  reduce  to 

a 

'=¥• 

In  order  to  interpret  this  new  result,  let  us  go  back  to  the  enun- 
ciation, and  it  will  be  perceived  that  it  is  absolutely  impossible  to 
satisfy  it  for  any  finite  value  for  t ;  for  whatever  time  we  allow  to 
the  two  couriers  they  can  never  come  together,  since  being  once  se- 
parated by  an  interval  a,  and  travelling  equally  fast,  this  interval 
will  always  be  preserved. 

Hence,  the  result,     —     may  be  regarded  as  a  sign  of  impossi- 

bility  for  any  finite  value  of  t. 


112  ALGEBRA. 

Nevertheless,  algebraists  consider  the  result 
a 

'=¥' 

as  forming  a  species  of  value,  to  which  they  have  given  the  name 
of  inJinHe  valve,  for  this  reason  : 

When  the  difference  m  —  n,  without  being  absolutely  nothing,  is 
supposed  to  be  very  small,  the  result 
a 


IS  ver}^  great. 

Take,  for  example,  ??«— n=0,01. 

^  a  a 

Then  i= =— — -=100a; 

7n~n       0,01 

Again,  take  ?«  —  ?i=r  0,001,  and  we  have 

In  short,  if  the  difference  between  the  rates  is  not  zero,  the  cou- 
riers will  come  together  at  some  point  of  the  line,  and  the  time  will 
become  greater  and  greater  as  this  difference  is  diminished. 

Hence,  if  the  difference  between  the  rates  is  less  than  any  assigna. 
hie  number,  the  time  expressed  by 

•a  a 

m—  n        0 ' 
iiill  he  greater  than  any  assignable  or  finite  number.     Therefore, 
for  brevity,  we  say  when  m— n=0,  the  result 
a 
~  m  —  n 
becomes  equal  to  infinity,  which  we  designate  by  the  character  oo. 
Again,  as  the  value  of  a  fraction  increases  as  its  numerator  be- 

A 
comes  greater  with  reference  to  its  denominator,  the  expression  — -, 

A  being  any  finite  number,  is  a  proper  symbol  to  represent  an  infi. 
nife  quantity  ;  that  is,  a  quantity  greater  than  any  assignable  quan- 
tity. 


EQUATIONS  OF  THE  FIRST  DEGREE.  113 

A  quantity  less  than  any  given  quantity  may  be  expressed  by  —  ; 
for  a  fraction  diminishes  as  its  denominator  becomes  greater  with 
reference  to  its  numerator.     Hence,  0  and  —  are  synonymous 

symbols,  and  so  are  —  and  ao  . 

We  have  been  thus  particular  in  explaining  these  ideas  of  infini- 
ty, because  there  are  some  questions  of  such  a  nature,  that  infinity 
may  be  considered  as  the  true  answer  to  the  enunciation. 

In  the  case,  just  considered,  where  m—n  it  will  be  perceived  that 
there  is  not,  properly  speaking,  any  solution  infinite  and  determinate 
numbers ;  but  the  value  of  the  unknown  quantity  is  found  to  be 
infinite. 

110.  If,  in  addition  to  the  hypothesis  m=n,  we  suppose  that  a=0, 

we  have 

0 

To  interpret  this  result,  let  us  reconsider  the  enunciation,  where 
it  will  be  perceived,  that  if  the  two  couriers  travel  equally  fast,  and 
are  once  at  the  same  point,  they  ought  always  to  be  together,  and 
consequently  the  required  point  is  any  point  whatever  of  the  line 

0 

travelled  over.     Therefore, the  expression     -—     is  in  this  case,  the 

symbol  of  an  indeterminate  quantity. 

If  the  couriers  do  not  travel  equally  fast,  that  is,  if  m>,  or  m<Cn, 
and  a=0,  then  will  f=0. 

Indeed,  it  is  evident,  that  if  the  couriers  travel  at  different  rates, 
and  are  together  at  12  o'clock,  they  can  never  be  together  after- 
wards. 

The  preceding  suppositions  are  the  only  ones  that  lead  to  remark- 
able  results  ;  and  they  are  sufficient  to  show  to  beginners  the  man- 
ner  in  which  the  results  of  algebra  answer  to  all  the  circumstances 
of  the  enunciation  of  a  problem. 

10* 


114  ALGEBRA. 

111.  We  will  add  another  example  to  show,  that  the  expression 
0 
—    expresses,  generally,  an  indeterminate  quantity. 

1—x 

Take  the  expression,     . 


1-a; 

Now,  if  we  perform  the  division  the  quotient  will  be  1 ;  and  if  we 
make  x=  1,  there  will  result 

1  -x_  0  _ 

Let  us  next  take  the  expression 


1-x    ' 

If  we.  perform  the   division,  the   quotient  will  be    1+x;  then 
making  x=l,  the  expression  becomes 
1-a,^       0 


1—x         0 

1—x'        0 

In  like  manner     •     — =-—=3  when  a;=l. 

1— a;         0 

l-x"        0  ' 

and  ....         _      — -— =n  when  x=l.  (See  Art. 59). 

all  of  which  goes  to  show  that     —     is  the  symbol  of  an  indetermi- 
nate  quantity. 

112.  We  will  add  another  example  showing  the  value  of  the  ex- 

Take  the  equation  ax=h,  involving  one  unknown  quantity,  whence 
b 


1st.  If,  for  a  particular  supposition  made  with  reference  to  the 

given   quantities   of  the   question,  we   have   a=0,   there  results 

h_ 

X-  ^. 

Now  in  this  case  the  equation  becomes  Ox^=^>  and  evidently 


EQUATIONS  OF  THE  FIRST  DEGREE.  115 

cannot  be  satisfied  by  any  finite  value  of  x.  We  will  however  remark 

b 
that,  as  the  equation  can  be  put  under  the  form    — =0,  if  we  sub- 

b 

stitute  for  x,  numbers  greater  and  greater,    —    will  ditTer  less  and 

less  from  0,  and  the  equation  will  become  more  and  more  exact ;  so 

b 
that,  we  may  take  a  value  for  x  so  great  that  —   will  be  less  than 

b 

any  assignable  quantity,  or   — =0. 

It  is  in  consequence  of  this  that  algebraists  say,  that  infinity  satis- 
fies the  equation  in  this  case  ;  and  there  are  some  questions  for 
which  this  kind  of  result  forms  a  true  solution  ;  at  least,  it  is  certain 
that  the  equation  does  not  admit  of  a  solution  m  finite  numbers,  and 
this  is  all  that  we  wish  to  prove. 

2d.  If  we  have  a—Q,  b=0,  at  the  same  time,  the  value  of  x 

0 
takes  the  fonn  x=-7r- 

In  this  case,  the  equation  becomes  Oxa;=0,  and  every  finite  num- 
ber, positive  or  negative,  substituted  for  x,  will  satisfy  the  equation. 
Therefore  iJie  equation,  or  the  probkm  of  which  it  is  the  algebraic 
translation,  is  indeterminate. 

0 
113.  It  should  be  observed,  that  the  expression  — ,   does  not  al- 
ways indicate  an  indetcrmination,  it  frequently  indicates  only  the  exist, 
ence  of  a  common  factor  to  the  two  terms  of  the  fraction,  which  fac- 
tor becomes  nothing,  m  consequence  of  a  particular  hypothesis. 
For  example,  suppose  that  we  find  for  the  solution  of  a  problem, 

x=-^ — :r7r.     If,  in  this  formula,  a  is  made  equal  to  b,  there  results 
ci?—b^ 

0 
x=-. 

But  it  will  be  observed,  that  a^—¥  can  be  put  under  the  form 
{a-b)  {a^'+ab+b^),  (Art.  59),  and  that  a^-J^  is  equal  to  {a-b) 


116  ALGEBRA. 

(a+b),  therefore  the  vakie  of  ;^ becomes 

_{a-b)  (a^+aZ.-l-i2) 
'"-'-      {a-b)  (a  +  b)      • 
Now,  if  we  suppress  the  common  factor  (a  —  b),  before  making 

the  supposition  a=J,  the  value  of  x  becomes     a;= —. , 

'^  a-\-b 

3rt2  3a 

which  reduces  to  a;=— - — ,  or  x=--,  when  a^Z*. 
2a  2 

For  another  example,  take  the  expression 

a2_J2   _  (a  +  b)  (a-h) 

{a-by~  (a-b)  {a-b}' 

0 

Making  a=b,  we  find  ^^='7r>  because  the  factor  (a  —  b)  is  com- 
mon to  the  two  terms ;  but  if  we  first  suppress  this  factor,  there  re- 

a+b       1  .  ,       1         ,  2« 

: 5-5  which  reduces  to  x=-—, 

a—b  0 

0 

From  this  we  conclude,  that  the  symbol     —    someUmes  indicates 

the  existence  of  a  common  factor  to  the  two  terms  of  the  fraction 
■which  reduces  to  this  form.  Therefore,  before  pronouncing  upon 
the  true  value  of  the  fraction,  it  is  necessary  to  ascertain  whether 
the  two  terms  do  not  contain  a  common  factor.  If  they  do  not,  we 
conclude  that  the  equation  is  really  indeterminate.  If  they  do  con- 
tain one,  suppress  it,  and  then  make  the  particular  hypothesis  ;  this 
will  give  the  true  value  of  the  fraction,  which  will  assume  one  of 

A     A      0 
the  three  forms  -^,  — -,  --,  in  which  case,  the  equation  is  determi- 

JD        0       0 

note,  impossible  in  finite  numbers,  or  indeterminate. 

This  observation  is  very  useful  in  the  discussion  of  problems. 

Of  Inequalities. 

114.  In  the  discussion  of  problems,  we  have  often  occasion  to 
suppose  several  inequalities,  and  to  perform  transformations  upon 
them,  analogous  to  those  executed  upon  equalities.     We  are  often 


INEQUALITIES.  117 

obliged  to  do  this,  when,  in  discussing  a  problem,  we  wish  to  esta- 
blish the  necessary  relations  between  the  given  quantities,  in  order 
that  the  problem  may  be  susceptible  of  a  direct,  or  at  least  a  real 
solution,  and  to  fix,  with  the  aid  of  these  relations,  the  limits  between 
which  the  particular  values  of  certain  given  quantities  must  be 
found,  in  order  that  the  enunciation  may  fulfil  a  particular  condition. 
Now,  although  the  principles  established  for  equations  are  in  general 
applicable  to  inequalities,  there  are  nevertheless  some  exceptions,  of 
which  it  is  necessary  to  speak,  in  order  to  put  the  beginner  upon  his 
guard  against  some  errors  that  he  might  commit,  in  making  use  of 
the  sign  of  inequality.  These  exceptions  arise  from  the  introduction 
of  negative  expressions  into  the  calculus,  as  quantities. 

In  order  that  we  may  be  clearly  understood,  we  will  take  exar 
pies  of  the  different  transformations  that  inequalities  may  be  subject- 
ed  to,  taking  care  to  point  out  the  exceptions  to  which  these  trans- 
formations are  liable. 

115.  Two  inequalities  are  said  to  subsist  in  the  same  sense,  when 
the  greater  quantity  stands  at  the  left  in  both,  or  at  the  right  in 
both  ;  and  m  a  contrary  sense,  when  the  greater  quantity  stands  at 
the  right  in  one,  and  at  the  left  in  the  other. 

Thus,     25>20     and     18>10,     or     6<8     and     7<9, 
ai'e  inequalities  which  subsist  in  the  same  sense  ;  and  the  inequalities 
15>13     and     12<14,     subsist  in  a  contrary  sense. 

1.  If  we  add  the  same  quantity  to  loth  members  of  an  inequality, 
or  subtract  the  same  quantity  from  both  members,  the  resulting  in- 
equality  will  subsist  in  the  same  sense. 

Thus,  take  8>6  ;  by  adding  5,  we  still  have  8  +  5>6  +  5 
and  8  — 5>6  — 5. 

When  the  two  members  of  an  equality  are  both  negative,  that 
one  is  the  least,  algebraically  considered,  which  contains  the  great- 
est number  of  units.  Thus,  —  25<  — 20  ;  and  if  30  be  added  to 
both  members,- we  have  5<10.  This  must  be  understood  entirely 
in  an  algebraic  sense,  and  arises  from  the  convention  before  esta- 


118  ALGEBRA. 

blished,  to  consider  all  quantities  preceded  by  the  minus  sign,  as 
subtractive. 

The  principle  first  enunciated,  serves  to  transpose  certain  terms 
from  one  member  of  the  inequality  to  the  other.     Take,  for  ex- 
ample,  the  inequality     a^  +  h''y2b-  —  2a^  ;    there  will  result  from  it 
a^+2a-y2h--P,  or  3a2>2^. 

2.  If  two  inequalities  subsist  in  the  same  sense,  and  tee  add  them 
member  to  member,  the  resulting  inequality  will  also  subsist  in  the  same 
sense. 

Thus,  from     ayb,cyd,eyf,   there  results    a-\-c-{-eyb-\-d-\-f 

But  this  is  not  always  the  case,  when  ice  subtract,  member  from  mem-  , 
her,  two  inequalities  established  in  the  same  sense. 

Let  there  be  the  two  inequalities  4<7  and  2<3,  we  have 
4-2  or  2<7-3  or  4. 

But  if  we  have  the  inequalities  9<10  and  6<8,  by  subtracting 
we  have     9—6  or  3>10  — 8  or  2. 

We  should  then  avoid  this  transformation  as  much  as  possible,  or 
if  we  employ  it,  determine  in  what  sense  the  resulting  inequality 
exists. 

3.  If  the  two  members  of  an  inequality  be  multiplied  by  a  positive 
number,  the  resulting  inequality  will  exist  in  the  same  sense. 

Thus,  from  a<ib,  we  deduce  3a<3J;  and  from  —a-C—k 
—  3a<  — 33. 

This  principle  serves  to  make  the  denominators  disappear. 

From  the  inequality    — —. — > — ,  we  deduce,  by  multiply. 

ing  by  6ad, 

M{a^-V)y2d{c''-d-). 

The  same  principle  is  true  for  division. 

But  when  the  two  members  of  an  inequality  are  multiplied  or  di- 
vided hy  a  negative  number,  the  inequality  subsists  in  a  contrary 
seme. 

Take,  for  example,  8>7;  multiplying  by  -3,  we  have 
-24<-21. 


INEQUALITIES.  119 

8  8  7 

In  like  manner,  8>7  gives   — — ,  or  — — <  — ^- 
—  3  3  3 

Therefore,  when  the  two  members  of  an  inequality  are  multipli- 

ed  or  divided  by  a  number  expressed  algebraically,  it  is  necessary 

to  ascertain  whether  the  multiplier  or  divisor  is  negative ;  for,  in 

that   case,    the  inequality  would  exist  in  a  contrary  sense. 

4.  It  is  not  permitted  to  change  the  signs  of  the  two  members  of  an 
inequality  unless  we  eslahlish  the  resulting  inequality  in  a  contrary 
sense ;  for  this  transformation  is  evidently  the  same  as  multiplying 
the  two  members  by   —1. 

5.  Both  memlers  of  an  inequality  between  positive  numbers  can  be 
squared,  and  the  inequality  loill  exist  in  the  same  sense. 

Thus  from  5>3,  we  deduce  25>9;    from   a+3>c,   we  find 

6.  Whe7i  both  members  of  the  inequality  are  not  positive,  we  cannot 
tell  before  the  operation  is  performed,  in  which  sense  the  resulting  in- 
equality will  exist. 

For  example,  — 2<3  gives  (  —  2)-  or  4<9;  but  — 3>— 5 
gives,  on  the  contrary,  (  — 3)^  or  9<(  — 5)-  or  25. 

We  must  then,  before  squaring,  ascertam  whether  the  two  mem- 
bers can  be  considered  as  positive  numbers. 

EXAMPLES. 

1.  Find  the  limit  of  the  value  of  x  in  the  expression 

5a;— 6>1.9.  Ans.     x>5. 

2.  Find  the  limit  of  the  value  of  x  in  the  expression 

14 

3x+— a;-30>10  Ans.     x>4. 

3.  Find  the  limit  of  the  value  of  x  in  the  expression 

1         1         a;      13     17 

T'^-Y'^+"2+'2">Y-  ^'^'    ^>^- 

4.  Find  the  limit  of  the  value  of  a;  in  the  inequalities 


120  ALGEBRA. 

ax  a? 

—  ^bx—ab^—. 

Ix  y 

-^-axArab<r-. 

5.  The  double  of  a  number  diminished  by  5  is  greater  than  25, 
and  triple  the  number  diminished  by  7,  is  less  than  double  the  num- 
ber increased  by  13.  Required  a  number  which  shall  satisfy  the 
conditions. 

By  the  question,  we  have 

2a;— 5>25. 
3x— 7<2x  +  13. 
Resolving  these  inequalities,  we  have  a;>15  and  a; < 20.     Any 
number,  therefore,  either  entire  or  fractional,  comprised  between  15 
and  20,  will  satisfy  the  conditions. 

6.  A  boy  being  asked  how  many  apples  he  had  in  his  basket,  re- 
plied,  that  the  sum  of  3  times  the  number  plus  half  the  number,  di- 
minished by  5  is  greater  than  16  ;  and  twice  the  number  diminished 
by  one  third  of  the  number,  plus  2  is  less  than  22.  Required  the 
number  which  he  had. 

Ans.     7,  8,  9,  10,  or  11. 


CHAPTER  III. 


Extraction  of  the  Square  Root  of  Numbers.  Forma- 
tion  of  the  Square  and  Extraction  of  the  Square 
Root  of  Algebraic  Quantities.  Calculus  of  Radi- 
cals of  the  Second  Degree.  Equations  of  the  Se- 
cond Degree. 

116.  The  square  or  second  power  of  a  number,  is  the  product 
which  arises  from  multiplying  that  number  by  itself  once  :  for  ex- 
ample,  49  is  the  square  of  7,  and  144  is  the  square  of  12. 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.  121 

The  square  root  of  a  number  is  a  second  number  of  such  a  value, 
that,  when  multiplied  by  itself  once  the  product  is  equal  to  the  given 
number.  Thus,  7  is  the  square  root  of  49,  and  12  the  square  root 
of  144:  for  7x^  =  49,  and  12x12  =  144. 

The  square  of  a  number,  either  entire  or  fractional,  is  easily 
found,  being  always  obtained  by  multiplying  this  number  by  itself 
once.  The  extraction  of  the  square  root  of  a  number,  is  however, 
attended  with  some  difficulty,  and  requires  particular  explanation. 

The  first  ten  numbers  are, 

1,     2,     3,       4,       5,       6,       7,       8,       9,       10, 
and  their  squares, 

.       1,     4,     9,     16,     25,     36,     49,     64,     81,     100. 
and  reciprocally,  the  numbers  of  the  first  line  are  tlie  square  roots 
of  the  corresponding  numbers  of  the  second.     We  may  also  remark 
that,  the  square  of  a  number  expressed  by  a  sihgle  figure,  will  contain 
no  figure  of  a  higher  denomination  than  tens. 

The  numbers  of  the  last  line  1,  4,  9,  16,  &c.,  and  all  other  num- 
bers which  can  be  produced  by  the  multiplication  of  a  number  by 
itself,  are  called  perfect  squares. 

It  is  obvious,  that  there  are  but  nine  perfect  squares  among  all  the 
numbers  which  can  be  expressed  by  one  or  two  figures  :  the 
square  roots  of  all  other  numbers  expressed  by  one  or  two  figures 
will  be  found  between  two  whole  numbers  differing  from  each  other 
by  unity.  Thus,  55  which  is  comprised  between  49  and  64,  has  for 
its  square  root  a  number  between  7  and  8.  Also,  91  which  is 
comprised  between  81  and  100,  has  for  its  square  root  a  number 
between  9  and  10. 

Every  number  may  be  regarded  as  made  up  of  a  certain  number 
of  tens  and  a  certain  number  of  units.  Thus  64  is  made  up  of  6 
tens  and  4  units,  and  may  be  expressed  under  the  form  60+4=64. 

Now,  if  we  represent  the  tens  by  a  and  the  units  by  b,  we  shall 
have  a-\-b    =  64 

and (a+&)2=(64f 

or      .     .     .       a-  +  2ai+Z>2  =4096. 
11 


122  ALGEBRA. 

Which  proves  that  the  square  of  a  number  composed  of  tens  and 
units  contains,  the  square  of  the  tens  plus  twice  the  product  of  the  tens 
by  the  units,  plus  the  square  of  the  units. 

117.  If  now,  we  make  the  units  1,  2,  3,  4,  &c.,  tens,  by  annex- 
ing to  each  figure  a  cipher,  we  shall  have, 

10,     20,     30,       40,       50,       60,       70,       80,       90,       100 
and  for  their  squares, 

100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000. 
from  which  we  see  that  the  square  of  one  ten  is  100,  the  square  of 
two  tens  400  ;  and  generally,  that  the  square  of  tens  will  contain  no 
figure  of  a  less  denoininaiion  than  hundreds,  nor  of  a  higher  name 
than  thousands. 

Example  I — To  extract  the  square  root  of  6084. 

Since  this  number  is  composed  of  more  than  two 
places  of  figures  its  roots  will  contain  more  than  one.  60.84 

But  since  it  is  less  than  10000,  which  is  the  square 
of  100,  the  root  will  contain  but  two  figures  :  that  is,  units  and  tens. 

Now,  the  square  of  the  tens  must  be  found  in  the  two  left  hand 
figures  which  we  will  separate  from  the  other  two  by  a  point. 
These  parts,  of  two  figures  each,  are  called  periods.  The  part  60 
is  comprised  between  the  two  squares  49  and  64,  of  which  the  roots 
are  7  and  8  :  hence,  7  is  the  figure  of  the  tens  sought ;  and  the  re- 
quired root  is  composed  of  7  tens  and  a  certain  number  of  units. 

The  figure  7  being  found,  we 
write   it  on  the  right  of  the  given  60.84  |  78 

number,  from  which  we  separate  it  49        I 

by  a  vertical  line  :  then  we  subtract  7x2  =  14.8  I  118.4 

its  square  49  from  60,  which  leaves  I  118  4 

a   remainder  of  11,  to  which  we  0 

bi'ing  down  the  two  next  figures  84. 

The  result  of  this  operation  1184,  contains  twice  the  product  of  the 
tens  by  the  units  plus  the  square  of  (he  units.  But  since  tens  multi- 
plied  by  units  cannot  give  a  product  of  a  less  name  than  tens,  it  fol- 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.    123 

lows  that  the  last  figure  4  can  form  no  part  of  the  double  product  of 
the  tens  by  the  units  :  this  double  product  is  therefore  found  in  the 
part  118,  which  we  separate  from  the  units'  place  4  by  a  point. 

Now  if  we  double  the  tens,  which  gives  14,  and  then  divide  118 
by  14,  the  quotient  8  is  the  figure  of  the  tmits,  or  a  figure  greater 
than  the  units.  This  quotient  figure  can  never  be  too  small,  since 
the  part  118  will  be  at  least  equal  to  twice  the  product  of  the  tens 
by  the  units  :  but  it  may  be  too  large  ;  for  the  118  besides  the  dou- 
ble  product  of  the  tens  by  the  units,  may  likewise  contain  tens  aris- 
ing  from  the  square  of  the  units.  To  ascertain  if  the  quotient  8 
expresses  the  units,  we  write  the  8  to  the  right  of  the  14,  which  gives 
148,  and  then  we  multiply  148  by  8.  Thus,  we  evidently  form, 
1st,  the  square  of  the  units  :  and  2d,  the  double  product  of  the  tens 
by  the  units.  This  multiplication  being  effected,  gives  for  a  product 
1184,  a  number  equal  to  the  result  of  the  first  operation.  Having 
subtracted  the  product,  we  find  the  remainder  equal  to  0  :  hence  78 
is  the  root  required. 

Indeed,  in  the  operations,  we  have  merely  subtracted  from  the 
given  number  6084,  1st,  the  square  of  7  tens  or  70 ;  2d,  twice  the 
product  of  70  by  8 ;  and  Sd,  the  square  of  8 :  that  is,  the  three 
parts  which  enter  into  the  composition  of  the  square  of  70  +  8  or 
78  ;  and  since  the  result  of  the  subtraction  is  0,  it  follows  that  78 
is  the  square  root  of  6084. 

Ex.  2.  To  extract  the  square  root  of  841. 
We  first  separate  the  number  into 
periods,  as  in  the  last  example.    In  the  8.41  I  29 

second  period,  which  contains  the  square  4        | 

of  the  tens,  there  is  but  one  figuz-e.  The  2x2=4.9  144.1 

greatest  square  contained  in  8  is  4,  the  I  44  1 

root  of  which  is  2  :  hence  2  is  the  fi-  0 

gure  of  the  tens  in  the  required  root. 

Subtracting  its  square  4  from  8,  and  bringing  down  41,  we  obtain 
for  a  result  441. 


124  ALGEBRA. 

If  now,  as  in  the  last  example,  we  separate  the  last  figure  1  from 
the  others  by  a  pohit,  and  divide  44  by  4,  which  is  double  the  tens, 
the  quotient  figure  will  be  the  units,  or  a  figure  greater  than  the 
units.  Here  the  quotient  is  11 ;  but  it  is  plain  that  it  ought  not  to 
exceed  9,  for  if  it  could,  the  figure  of  the  tens  already  found  would 
be  too  small.  Let  us  then  try  9.  Placing  9  in  the  root,  and  also 
on  the  right  of  the  4,  and  multiplying  49  by  9,  we  obtain  for  a  pro- 
duct 441  :  hence,  29  is  the  square  root  of  841. 

Remark.  The  quotient  figure  11,  first  found,  was  too  large  be- 
cause the  dividend  44  contained,  besides  the  double  product  of  the 
tens  by  the  units,  8  tens  arising  from  the  square  of  the  units.  When 
the  dividend  is  considerably  augmented,  by  tens  arising  from  the 
square  of  the  units,  the  quotient  figure  will  be  too  large. 

Ex.  3.  To  extract  the  square  root  of  431649. 

Since  the  given  number  exceeds  10,000  its  root  will  be  greater 
than  100  ;  that  is,  it  will  contain  more  than  two  places  of  figures. 
But  we  may  still  regard  the  root  as  composed  of  tens  and  units,  for 
every  number  may  be  expressed  in  tens  and  units.  For  example, 
the  number  6758  is  equal  to  675  tens  and  8  units,  equal  to  6750  +  8. 

Now,  we  know  that  the  square  of  the  tens  of  the  required  root 
can  make  no  part  of  the  two  right 

hand  figures  49,  which  therefore,  we  43.16.49  |  657 

separate  from  the  others  by  a  point, 
and  the  remaining  figures  4316  con- 
tain the  square  of  the  tens  of  the  re- 
quired root.  But  since  4316  exceeds 
100  the  tens  of  the  required  root  will 
contain  more  than  one  figure  :  hence 
4316  must  be   separated   into    two 

parts,  of  which  the  right  hand  period  16  will  contain  no  part  of  the 
square  of  that  figure  of  the  root,  which  is  of  the  highest  name,  and 
for  a  similar  reason  we  should  separate  again  if  the  part  to  the  left 
contained  more  than  two  figures. 


43.16.49 

12.5 

36 
71.6 

5 

62  5 

130.7 

9  14.9 

9  14  9 

0 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.  125 

Since  36  is  the  greatest  square  contained  in  43,  the  first  figure  of 
the  root  is  6.  We  then  subtract  its  square  36  from  43,  and  to  the 
remainder  7  bring  down  the  next  period  16.  Now,  since  the  last 
figure  6  of  the  result  716,  contains  no  part  of  the  double  product  of 
the  first  figure  of  the  tens  by  the  second,  it  follows,  that  the  second 
figure  of  the  root  will  be  obtained  by  dividing  71  by  12,double  the 
first  figure  of  the  tens.  This  gives  5  for  a  quotient,  which  we  place 
in  the  root,  and  at  the  right  of  the  divisor  12.  Then  subtract  the 
product  of  125  by  5  from  716,  and  to  the  remainder  bruig  down  the 
next  period,  and  the  result  9149  will  contam  twice  the  product  of  the 
tens  of  the  root  multiplied  by  the  units,  plus  the  square  of  the  units. 
If  this  result  be  then  divided  by  twice  65,  that  is,  by  double  the  tens 
of  the  root,  (which  may  always  be  found  by  adding  the  last  figure 
of  the  divisor  to  itself),  the  quotient  will  be  the  units  of  the  root. 

Hence,  for  the  extraction  of  the  square  root  of  numbers,  we  have 
the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures  each  be- 
ginning at  the  right  hand, — the  period  on  the  left  will  often  contain  but 
one  figure. 

II.  Find  the  greatest  square  in  the  first  period  on  the  left,  and  place 
its  root  on  the  right  after  the  manner  of  a  quotient  in  division.  Sub- 
tract the  square  of  the  root  from  the  first  period,  and  to  the  remainder 
bring  down  the  second  period  for  a  dividend. 

III.  Double  the  root  already  found  and  place  it  on  the  left  for  a  di- 
visor. Seek  how  many  times  the  divisor  is  contained  in  the  dividend, 
exclusive  of  the  right  hand  figure,  and  place  the  figure  hi  the  root  and 
also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor,  thus  augmented,  by  the  last  figure  of  the 
root,  and  subtract  the  product  from  the  dividend,  and  to  the  remainder 
bring  doion  the  next  period  for  a  new  dividend. 

V.  Double  the  whole  root  already  found,  for  a  new  divisor,  and 
continue  the  operation  as  before,  until  all  the  periods  are  brought  doicn. 

11* 


126  ALGEBRA. 

1st.  Remark.  If,  after  all  the  periods  are  brought  down,  there  is 
no  remainder,  the  proposed  number  is  a  perfect  square.  But  if 
there  is  a  remainder,  you  have  only  found  the  root  of  the  greatest 
perfect  square  coutauied  m  the  given  number,  or  the  entire  part  of 
the  root  sought. 

For  example,  if  it  were  required  to  extract  the  square  root  of  665, 
we  should  find  25  for  the  entire  part  of  the  root  and  a  remainder  of 
40,  which  shows  that  665  is  not  a  perfect  square.  But  is  the  square 
of  25  the  greatest  perfect  square  contained  in  665  ?  that  is,  is  25  the 
entire  part  of  the  root  ?  To  prove  this,  we  will  first  show  that,  the 
difference  between  the  squares  of  two  conseciUive  numlers,  is  equal  to 
twice  the  less  numher  augmented  hy  unity. 

Let         .         .         .         a    =     the  less  number, 

and     .         .         .  a  +  1    =     the  greater. 
Then      .         .  (a+l)2=a''+2a+l 

and     .  .  .        {af=a- 

Their  difference  is   .     ■         =        2a+l     as  enunciated. 
Hence,  the  entire  part  of  the  root  cannot  be  augmented,  unless 
the  remainder  exceed  twice  the  root  found,  plus  unity. 

But  25  X  2+1 =51  >  40  the  remauider  :  therefore,  25  is  the  en- 
tire  part  of  the  root. 

2d.  Remark.  The  number  of  figures  m  the  root  will  always  be 
equal  to  the  number  of  periods  into  which  the  given  number  is 
separated. 

EXAMPLES. 

1.  To  find  the  square  root  of  7225. 

2.  To  find  the  square  root  of  17689. 

3.  To  find  the  square  root  of  994009. 

4.  To  find  the  square  root  of  85678973. 

5.  To  find  the  square  root  of  67812675. 

118.  The  square  root  of  a  number  which  is  not  a  perfect  square, 
is  called  incommensuraljle  or  irrational,  because  its  exact  root  can- 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.  127 

not  be  found  in  terms  of  the  numerical  unit.     Thus,     V^      Vs, 

V77     are  incommensurable  numbers.     They  are  also  sometimes 
called  surds. 

In  order  to  prove  that  the  root  of  an  imperfect  power  cannot  be 
expressed  by  exact  parts  of  unity,  we  must  first  show  that. 

Every  numher  P,  lohich  loill  exactly  divide  the  product  A  xB  of  two 
numbers,  and  which  is  prime  with  one  of  them,  will  divide  the  other. 
Let  us  suppose  that  P  will  not  divide  A,  and  that  A  is  greater 
than  P.  Apply  to  A  and  P  the  process  for  finding  the  greatest  com- 
mon divisor,  and  designate  the  quotients  which  arise  by  Q,  Q',  Q"  .  .  . 
and  the  remainders  R,  R',  R"  .  .  .  respectively.  If  the  division  be 
continued  sufficiently  far,  we  shall  obtain  a  remainder  equal  to  unity, 
for  the  remainder  cannot  be  0,  since  by  hypothesis  A  and  P  are  prime 
with  each  other.     Hence  we  shall  have  the  following  equations. 

A=P  Q    +R 

P  =R  Q'  +R' 

R=iR'Q"+R" 

R'=R"Q"'+R"' 


Multiplying  the  first  of  these  equations  by  B,  and  dividing  by  P, 

we  have 

AB        ^      BR 
— =BQ+—. 

AB  .  ,      .  r. 

But,  by  hypothesis,       „        is  an  entire  number,  and  since  B  and 

Q  are  entire  numbers,  the  product  BQ  is  an  entire  number.  Hence 

BR 

it  follows  that       „        is  an  entire  number. 

If  we  multiply  the  second  of  the  above  equations  by   B,  and 
divide  by  P,  we  have 

BRQ'       BR' 


128  ALGEBRA. 

RR 

But  we  have  already  shown  that  is  an  entire   number  ; 

B^Q'      •  •  .         mu-  .  •       ,  BR' 

hence     — 5 —     is  an  entire  number.      1  his  being  the  case,     -r^— 

must  also  be  an  entire  number.     If  the  operation  be  continued  until 

Bxl 

the  number  which  multiplies  B  becomes  1,  we  shall  have     — - — 

equal  to  an  entire  number,  which  proves  that  P  will  divide  B. 

In  the  operations  above  we  have  supposed  A>P,  but  if  P>  A  we 
should  first  divide  P  by  A. 

Hence,  if  a  number  P  xvill  exactly  divide  the  product  of  iico  num.- 
hers,  and  is  prime  with  one  of  them,  it  loill  divide  the  other. 

We  will  now  show  that  the  root  of  an  imperfect  power  cannot  be 
expressed  by  a  fractional  number. 

Let  c  be  an  imperfect  square.     Then  if  its  exact  root  can  be  ex- 
pressed  by  a  fractional  number,  we  shall  have 
/ —     « 


or     .  •         .  .         ^^="17  by  squaring  both  members. 

But  if  c  is  not  a  perfect  power,  its  root  will  not  be  a  whole  num- 
ber, hence     -7-     will  at  least  be  an  irreducible  fraction,  or  a  and  b 
0 

will  be  prime  to  each  other.     But  if  a  is  not  divisible  hj  b,  axo-  ox 

c?  will  not  be  divisible  by  b,  from  what  has  been  shown  above  ; 

neither  then  can  c^  be  divisible  by  ¥.     Smce  to  divide  by  ¥  is  but 

a^ 
to  divide  cP  twice  by  b.     Hence,     -j^     is  an  irreducible  fraction, 

and  therefore  cannot  be  equal  to  the  entire  number  c  :  therefore,  we 

. —  a 
cannot  assume      v  c—-j,     or  the  root  of  an  miperfect  power  can- 
not be  expressed  by  a  fractional  number  that  is  rational. 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS.         129 


Extraction  of  the  square  root  of  Fractions. 

119.  Since  the  square  or  second  power  of  a  fraction  is  obtained 
by  squaring  the  numerator  and  denominator  separately,  it  follows 
that  the  square  root  of  a  fraction  will  be  equal  to  the  square  root 
of  the  numerator  divided  by  the  square  root  of  the  denominator. 

a^  a 

For   exainple,    the   square    root    of      7^     is  equal  to    -r-  '•  ^or 


a       a 


b'^b      b' 

But  if  neither  the  numerator  nor  the  denominator  is  a  perfect 
square,  the  root  of  the  fraction  cannot  be  exactly  found.  We  can 
however,  easily  find  the  exact  root  to  within  less  than  one  of  the 
equal  parts  of  the  fraction. 

To  effect  this,  multiply  both  terms  of  the  fraction  by  the  denomina- 
tor, which  makes  the  denominator  a  perfect  square  without  altering  the 
value  of  the  fraction.  Then  extract  the  square  root  of  the  perfect 
square  nearest  the  value  of  the  numerator,  and  place  the  root  of  the 
denominator  under  it;  this  fraction  will  be  the  approximate  root. 

3 
Thus,  if  it  be  required  to  extract  the  square  root  of  — ,  wemul- 

15 

tiply  both  terms  by  5,  which  gives     —  :     the  square  nearest  15  is 

4 

16  :  hence    —     is  the  required  root,  and  is  exact  to  within  less 
o 

1 
than     — . 
5 

120.  We  may,  by  a  similar  method,  determine  approximatively 
the  roots  of  whole  numbers  which  are  not  perfect  squares.  Let  it 
be  required,  for  example,  to  determine  the  square  root  of  an  entire 

number  a,  nearer  than  the  fraction  — :    that  is  to  say,  to  find  a 


130  ALGEBRA. 

number  which  shall  differ  from  the  exact  root  of  a,  by  a  quantity 

less  than    — . 
n 

It  may  be  observed  that  a=:— j-.     If  we    designate  by   r   the 

entire  part  of  the  root  of  air,  the  number  an^  will  then  be  compris- 

ed  between  r^  and.  {r+Vf\  and     — —    will  be  comprised  between 


'     -"^ — j-^  ;     and  consequently  the  true  root  of  a  is  com- 


and 


prised  between  the  root  of    -^     and     5- —  ;     that   is,  between 

'  ir  n^ 

—     and     .     Hence     —     will  represent  the  square  root  of  a 

n  n  n 

within  less  than  the  fraction     — .     Hence  to  obtain  the  root : 
n 

Multiply  the  given  number  hij  the  square  of  the  denominator  of  the 
fraction  which  determines  the  degree  of  approximation  :  then  extract 
the  square  root  of  the  product  to  the  nearest  unit,  and  divide  this  root 
by  the  denominator  of  the  fraction. 

Suppose,  for  example,  it  were  required  to  extract  the  square  root 

1 
of  59,  to  within  less  than     — . 

Let  us  repeat  on  this  example,  the  demonstration  which  has  just 
been  made. 

59x(12f 
The  number  59  can  be  put  under  the  form    — — -^ — ,     or  by 

8496 
multipliying   by   (12)2,     —-.     But  the  root  of  8496  to  the  near- 

8496 
est  unit,  is  92  :  hence  it  follows  that    — -r^    or  59,  is  comprised  be- 


tween     7— ^     and     )— ^.     Then,  the  square  root  of  59  is  itself 
( 12 )  ( Iz) 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS.        131 
92  93 

comprised  between     —     and     — :     that  is  to  say,  the  true  root 

92  ,  1 

differs  from     —     by  a  fraction  less  than     — . 
12  I'* 

92  93  8464  8649 

Indeed  the  squares  of    —  and  —  are  ^  and  ,   num. 

8496 
bers  which  comprise  or  59. 

{i-'i) 

2.  To  find  the  VTT  to  within  less  than     t-t. 

15 


, 1 

3.  To  find  the   V223    to  within  less  than '   — . 


4 

Ans.     3—. 
15 


37 

Ans.     14-—. 

40 


121.  The  manner  of  determining  the  approximate  root  in  deci- 
mals, is  a  consequence  of  the  preceding  rule. 

.    .        1         1 

To  obtain  the  square  root  of  an  entire  number  within     — , 

r-T-r-r,  «Sjc. — it  is  neccssary  according  to  the  preceding  rule  to  mul- 
tiply the  proposed  number  by  (10)^  (100)^  (1000)^  .  .  .  or,  which 
is  the  same  thing,  add  to  the  riglit  of  the  number,  two,  four,  six,  ^c. 
ciphers :  then  extract  the  root  of  the  product  to  the  nearest  unit,  and 
divide  this  root  hy  10, 100, 1000,  &c.,  which  is  effected  hy  pointing  off 
one,  two,  three,  <f-c.,  decimal  places  frojn  the  right  hand. 

Example  1.  To  extract  the  square  root  of  7  to  within     ~T7r7r' 

Having  added   four  ciphers  to  the  7.00.00  I  2,64 

right  hand  of  7,  it  becomes  70000,  4              I 

whose  root  extracted  to  the  nearest  46  I  300 

unit  is  264,  which  being  divided  by  I  276 

100  gives  2,64  for  the  answer,  which  524  I     2400 

.,.    ,         ^  1  I     2096 

IS  true  to  within  less  than     -r— --.  — rr-,     „  • 

100  304     Rem 


132 

ALGEBRA. 

2. 

Find  the    ^29  to  within     -j— r- 

3. 

Find  the    V  227    to  within     -— - 

A71S.     5,38. 
1 
lOOOO"' 

Ans.     15,0665. 
Remark.     The  number  of  ciphers  to  be  annexed  to  the  whole 
number,  is  always  double  the  number  of  decimal  places  required  to 
be  found  in  the  root. 

122.  The  manner  of  extracting  the  square  root  of  decimal  frac- 
tions  is  deduced  immediately  from  the  preceding  article. 

Let  us  take  for  example  the  number  3,425.     This  fraction  is 

3425      ^  ■ 
equivalent  to     tt^^-     Now  1000  is  not  a  perfect  square,  but  the 

denominator  may  be  made  such  without  altering  the  value  of  the 

34250 

fraction,  by  multiplying  both  the  terms  by  10  ;  this  gives  ■■■ 

34250 
or  -^.     Then  extracting  the  square  root  of  34250  to  the 


185 

i  185  ;  hence     ■ 

1 


nearest  unit,  we  find  185 ;  hence  or  1,85  is  the  required 


root  to  within 

If  greater  exactness  be  required,  it  will  be  necessary  to  add  to 
the  number  3,4250  so  many  ciphers  as  shall  make  the  periods  of 
decimals  equal  to  the  number  of  decimal  places  to  be  found  in  the 
root.     Hence,  to  extract  the  square  root  of  a  decimal  fraction  : 

Annex  ciphers  to  the  proposed  numler  until  the  decimal  places  shall 
he  even,  and  equal  to  double  the  number  of  pilaces  required  in  the  root. 
Then  extract  tlie  root  to  the  nearest  unit,  and  point  off  from  the  right 
hand  the  required  number  of  decimal  places 


Ex.   1.  Find  the      V  3271,4707     to  within     ,01. 

Ans.     57,19. 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.  133 

2.  Find  the      V  31,027     to  within     ,01. 

Ans.     5,57. 


3.  Find  the      Vo,01001     to  within     ,00001. 

Ans.     0,10004. 

123.  Finally,  if  it  be  required  to  find  the  square  root  of  a  vulgar 
fraction  in  terms  of  decimals  :  Change  the  vulgar  fraction  into  a  de- 
cimal and  continue  the  division  until  the  number  of  decimal  places  is 
double  the  number  of  places  required  in  the  root.  Then  extract  the 
root  of  the  decimal  by  the  last  rule. 

11 

Ex.  1.  Extract  the  square  root  of    —     to  within  ,001.     This 

number,  reduced   to   decimals,  is  0,785714  to  within  0,000001  ; 
but  the  root  of  0,785714  to  the  nearest  unit,  is  ,886  :  hence  0,886  is 

the  root  of    — •     to  within  ,001. 
14 


*  /     13 

2.  Find  the     'V    2 -—     to  within  0,0001. 
15 


Ans.     1,6931. 


Extraction  of  the  Square  Root  of  Algebraic  Quantities. 

124.  We  will  first  consider  the  case  of  a  monomial ;  and  in  order 
to  discover  the  process,  see  how  the  square  of  the  monomial  is 
formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  41.),  we 
have 

{ba''}Pcy=hd'Wc  X  MWc=  Iha^Wc^  ; 
that  is,  in  order  to  square  a  monomial,  it  is  necessary  to  square  its 
co-efflcient,  and  double  each  of  the  exponents  of  the  different  letters. 
Hence,  to  find  the  root  of  the  square  of  a  monomial,  it  is  necessary, 
1st.  To  extract  the  square  root  of  the  co-efcient,  2d.  To  take  the 
half  of  each  of  the  exponents. 

Thus,      V6Wb*^8a''l^  ',  for  8a^'^^'«^'='^>2_64a«J*. 
12 


134 

In  like  manner, 


-v/625a^*V=25a&*c",  for  {25ah'cy=62ba^h\ 

From  the  preceding  rule,  it  follows,  that,  when  a  monomial  is  a 
perfect  square,  its  numerical  co-efficient  is  a  perfect  square,  and  all 
its  exponents  even  nunibers.  Thus,  25a^Z>^  is  a  perfect  square,  but 
98a&*  is  not  a  perfect  square,  because  98  is  not  a  perfect  square,  and 
a  is  affected  with  an  uneven  exponent. 

In  the  latter  case,  the  quantity  is  introduced  into  the  calculus  by 
affecting  it  with  the  sign  V  ,  and  it  is  written  thus,  VdSah*. 
Quantities  of  this^  kind  are  called  radical  quantities,  or  irrational 
qvMntities,  or  simply  radicals  of  the  second  degree. 

125.  These  expressions  may  sometimes  be  simplified,  upon  the 
principle  that,  the  square  root  of  the  product  of  two  or  more  factors  is 
equal  to  the  product  of  the  square  roots  of  these  factors ;  or,  in  alge- 
braic language,      Vabcd  .  .  .  =  y/a.  y/h.  \/c.  ^/d.  .  .  . 

To  demonstrate  this  principle,  we  will  observe,  that  from  the  de- 
finition of  the  square  root,  we  have 

(  Valcd  .  .  .  .f=abcd  .... 

Again, 
(v/aX  v/^X  v/cX  W  .  .  f=(%/«)'x(%/Jfx(v/c;)'x(  W)'  •  •  • 
=zal)cd  .... 

Hence,  since  the  squares  of      Vabcd  .  .  .  .,  and, 

-v/a.  ■s/b.  -v/c.  \/d.  .  .  .,  are  equal,  the   quantities  themselves   are 

equal.  

This  being  the  case,  the  above  expression,    ■V98ab*,  can  be  put 

under  the  form    VA9b'x2a=  V^9^X  V2a.     Now  Vio^may 
be  reduced  to  IP  ;  hence   VoSab^z^lb^  V2a. 
In  like  manner, 


V^So^ZiVd^  V9a-lr'c''x5bd  =  Sabc  V5bd, 
Vse4:aWc''=  VTUa^b^><Mc=l2aif^c^  Vdbc. 
The  quantity  which  stands  without  the  radical  sign  is  called  the 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.        135 

co.efficient   of  the  radical.     Thus,   m.   the   expressions    W  V^a, 

2aic  Vbbd,     I2a¥c^  VqIc,    the  quantities  Itf^,  3ahc,  I2alr^c^,  are 
called  co-efficients  of  the  radicals. 

In  general,  to  simplify  an  irrational  monomial,  separate  it  into 
two  parts,  of  lohich  one  shall  contain  all  tlie  factors  that  are  perfect 
squares,  and  the  other  the  remaining  ones :  then  take  the  roots  of  the 
perfect  squares  and  place  them  before  the  radical  sign,  under  which, 
leave  those  factors  which  are  not  perfect  squares. 

EXAMPLES. 


1.  To  reduce      Vlbd^bc     to  its  simplest  form. 

2.  To  reduce      V\2Q¥aH"     to  its  simplest  foi-m. 

3.  To  reduce      v  32a^i^c       to  its  simplest  form. 

4.  To  reduce      V^bQa^b'^c^     to  its  simplest  form. 

5.  To  reduce      Vl024a''^V     to  its  simplest  form. 

6.  To  reduce      Vl'lQa'Wc'^d     to  its  simplest  form. 

126.  Since  like  signs  in  both  the  factors  give  a  plus  sign  in  the 
product,  the  square  of  —a,  as  well  as  that  of  +«,  will  be  a^ : 
hence  the  root  of  a-  is  either  +a  or  —a.  Also,  the  square  root 
of  25a^Z>^  is  either  +baV^  or  —ha¥.  Whence  we  may  conclude, 
that  if  a  monomial  is  positive,  its  square  root  may  be  affected  either 
with  the  sign  +  or  —  ;  thus,  VdcF^±2a",  for  +3a2  or  —Za^, 
squared,  gives  9a*.  The  double  sign  ±  with  which  the  root  is 
affected  is  read  plus  or  minus. 

If  the  proposed  monomial  were  negative,  it  would  be  impossible 
to  extract  its  root,  since  it  has  just  been  shown  that  the  square  of 
every  quantity,  whether  positive  or  negative,  is  essentially  positive. 
Therefore,  V— 9,  V— 4a^  V  —  Sa%  are  algebraic  symbols 
which  indicate  operations  that  cannot  be  performed.  They  are 
called  imaginary  quantities,  or  rather  imaginary  expressions,  and  are 
frequently  met  with  in  the  resolution  of  equations  of  the  second 


136  ALGEBRA. 

degree.     These  symbols  can,  however,  by  extending  the  rules,  be 
simplified  in  the  same  manner  as  those  irrational  expressions  which 

indicate  operations  that  cannot  be  performed.  Thus,  V  — 9  may  be 
reduced  to  (Art.  125.) 
Vox   V^7or,3  V-l;   V-Aa^^  VlaFx  V-l  =  2a  \/^ 

127.  Let  us  now  examine  the  law  of  formation  for  the  square  of 
any  polynomial  whatever ;  for,  from  this  law,  a  rule  is  to  be  de- 
duced for  extracting  the  square  root. 

It  has  already  been  shown  that  the  square  of  a  binomial  (a  +  b) 
is  equal  to  a^-\-2ah+P  (Art.  46.). 

Now  to  form  the  square  of  a  trinomial  a-{-b-{-c,  denote  a +3  by 
the  single  letter  s,  and  we  have 

(a+J+c)^=(5+c)-=:^+25c+c2. 

Bui  s^^(a+bY^a^  +  2ab+Ir' ;  and  2sc=2{a+b)c=2ac-]-2bc. 

Hence     (a+b+cy=a"+2ab  +  I^+2ac+2bc+c^ ; 
that  is,  tJie  square  of  a  trinomial  is  composed  of  the  sum  of  the  squares 
of  its  three  terms,  and  twice  the  products  of  these  terms  multiplied 
together  two  and  two. 

If  we  take  a  polynomial  of  four  or  more  terms,  and  square  it,  we 
shall  find  the  same  law  of  formation.  We  may,  therefore,  suppose 
the  law  to  be  proved  for  the  square  of  a  polynomial  of  m  terms  ; 
and  it  then  only  remains  to  be  shown  that  it  will  be  true  for  a  poly- 
nomial  of  m+1  terms. 

Take  the  polynomial  {a-\-b+c  .  .  .  +?),  having  m  terms,  and 
denote  their  sum  by  s\  then  the  polynomial  (a-\-b-{-c  .  .  .  +i+k) 
having  ??i  +  l  terms,  will  be  denoted  by  (*+/t). 

Now,         {s-{-kf=^+2sk-]-k~,  or  by  substituting  for  s. 
{s-\-kf=(a+b-{-c  .  .  .  +iy+2{a+b+c  .  .  .-?-i)k+k''. 

But  by  hypothesis,  the  first  part  of  this  expression  is  composed 
of  the  squares  of  all  the  terms  of  the  first  polynomial  and  the  double 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.       137 

products  of  these  terms  taken  two  and  two  ;  the  second  part  contains 
the  double  products  of  all  the  terms  of  the  first  polynomial  by  the 
additional  term  Tc ;  and  the  third  part  is  the  square  of  this  term. 
Therefore,  the  law  of  composition,  announced  above,  is  true  for  the 
new  polynomial.  But  it  has  been  proved  to  be  true  for  a  trinomial ; 
hence  it  is  true  for  a  polynomial  containing  four  terms  ;  being  true 
iovfour,  it  is  necessarily  true  for^t-e,  and  so  on.  Therefore  it  is 
general.     This  law  can  be  enunciated  in  another  manner  :  viz. 

The  square  of  any  polynomial  contains  the  square  of  tJiefrtt  term, 
plus  twice  the  product  of  the  first  hy  the  second,  plus  the  square  of  the 
second ;  plus  twice  the  product  of  tfie  first  two  terms  ly  the  third,  plus 
the  square  of  the  third ;  plus  twice  the  product  of  the  first  three  terms 
by  the  fourth,  plus  the  sqiuire  of  the  fourth  ;  and  so  on. 

This  enunciation  which  is  evidently  comprehended  in  the  first, 
shows  more  clearly  the  process  for  extracting  the  square  root  of  a 
polynomial. 

From  this  law, 

(^a+i+cf=a'^+2ab-\-lr+2{a-^h)c+c^ 

{a+b+c+df=d'+2ab-{-lr'+2{a  +  b)c  +  c''+2{a  +  b+c)d+d^. 

128.  We  will  now  proceed  to  extract  the  square  root  of  a  poly- 
nomial. 

Let  the  proposed  polynomial  be  designated  by  N,  and  its  root, 
which  we  wUl  suppose  is  determined,  by  R  ;  conceive,  also,  that 
these  two  polynomials  are  arranged  with  reference  to  one  of  the 
letters  which  they  contain,  a,  for  example. 

Now  it  is  plain  that  the  first  term  of  the  root  R  may  be  found  by 
extracting  the  root  of  the  first  term  of  the  polynomial  N  ;  and  that 
the  second  term  of  the  root  may  be  found  by  dividing  the  second 
term  of  the  polynomial  N,  by  twice  the  first  term  of  the  root  R. 

If  now  we  form  the  square  of  the  binomial  thus  found,  and  sub- 
tract  it  from  N,  the  first  term  of  the  remainder  will  be  twice  the 
product  of  the  first  term  of  R  by  the  third  term  :  hence,  if  this  first 
12* 


138  ALGEBRA. 

term  be  divided  by  double  the  first  term  of  R,  the  quotient  will  be 
the  third  term  of  R. 

In  order  to  obtain  the  fourth  term  of  R,  form  the  double  products 
of  the  first  and  second  terms,  by  the  third,  plus  the  square  of  the 
third ;  then  subtract  all  these  products  from  the  remainder  before 
found,  and  the  first  term  of  the  result  will  be  twice  the  product  of 
the  first  term  of  the  root  by  the  4th  :  hence,  if  it  be  divided  by 
double  the  first  term,  the  quotient  will  be  the  fourth  term.  In  the 
same  manner  the  next  and  subsequent  terms  may  be  found.  Hence, 
for  the  extraction  of  the  square  root  of  a  polynomial  we  have  the 
following 

RULE. 

I.  Arrange  the  polynomial  with  reference  to  one  of  its  letters  and 
extract  the  square  root  of  the  first  term  :  this  will  give  the  first  term 
of  the  root. 

II.  Divide  the  second  term  of  the  polynomial  by  double  the  first 
term,  of  ilie  root,  and  tlie  quotient  will  be  the  second  term  of  the  root. 

III.  Then  form  the  square  of  the  two  terms  of  the  root  found,  and 
subtract  it  from  the  first  polynomial,  and  then  divide  the  first  term  of 
the  remainder  by  double  the  first  term  of  the  root,  and  the  quotient 
vnll  he  the  third  term. 

IV.  Form  the  double  products  of  the  first  and  second  terms,  by  the 
third,  plus  the  square  of  the  third ;  then  subtract  all  these  products 
from  the  last  remaiJider,  and  divide  the  first  term  of  the  result  by  dou- 
ble tJie  first  term  of  the  root,  and  the  quotient  toill  be  the  fourth  term. 
Then  proceed  in  the  same  manner  to  find  the  other  terms. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial 

49a^Ir'-24:aP+25a''-S0a''b  +  16b*. 
First  arrange  it  with  reference  to  the  letter  a. 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.       139 


25a*-S0a'b-{-A9a^--24aP+16b* 

5a'  —  Sab+4,P 

25a*— 30a^5+   Qa^^ 

lOa^ 

4:0a-b'-24.aP+16M 

1st.  Rem. 

^0aW-24aP  +  16¥ 

0  .  .  .  2cl.  Rem. 
After  having  arranged  the  polynomial  with  reference  to  a,  extract 
the  square  root  of  SSa",  this  gives  5a^,  which  is  placed  to  the  right 
of  the  polynomial;  then  divide  the  second  term,  —SOa%  by  the 
double  of  5a^,  or  lOa^;  the  quotient  is  —3ab,  and  is  placed  to  the 
right  of  5a^.  Hence,  the  first  two  terms  of  the  root  are  5a^—3ab. 
Squaring  this  binomial,  it  becomes  25a*~30a^i+9a^Z^,  which,  sub- 
tracted  from  the  proposed  polynomial,  gives  a  remainder,  of  which 
the  first  term  is  AOaW.  Dividing  this  first  term  by  lOa^  (the  double 
of  5a^),  the  quotient  is  +4^- ;  this  is  the  third  term  of  the  root,  and 
is  written  on  the  right  of  the  first  two  terms.  Forming  the  double 
product  of  5a^—3ab  by  4J^,  and  the  square  of  45^,  we  find  the  poly- 
nomial 40a^^— 24aZ'^+16Z>*,  which,  subtracted  from  the  first  re- 
mainder,  gives  0.     Therefore  5a^—Sab-i-4P  is  the  required  root. 

2.  Find  the  square  root  of 

a*  +  'ia\v + ea^r" + 4fl.r^ + x*. 

3.  Find  the  square  root  of 

a*-2a='a,'+3aV-2aar'+«*. 

4.  Find  the  square  root  of 

Ax^  +  l2x^  +  5x*-2x^+7x^-2x+l. 

5.  Find  the  square  root  of 

9a*-12a'b+28a'P-16aP+16b*. 

6.  Find  the  square  root  of 

25a*  J2_  40a^52c + 760=^^0^- 48aJ2c='+ 36  J^c*  -  30a*3c + 24a33c^ 

-36a25c='+9aV. 
189.  We  will  conclude  this  subject  with  the  following  remarks. 
1st.  A  binomial  can  never  be  a  perfect  square,  smce  we  know 
that  the  square  of  the  most  simple  polynomial,  viz.  a  binomial,  con- 


140  ALGEBRA. 

tains  three  distinct  parts,  Mhich  cannot  experience  any  reduction 
amongst  themselves.  Thus,  the  expression  ar-\-¥  is  not  a  perfect 
square  ;  it  wants  the  term  ±2a5  in  order  that  it  should  be  the  square 
of  aztil). 

2d.  In  order  that  a  trinomial,  when  arranged,  may  be  a  perfect 
square,  its  two  extreme  terms  must  be  squares,  and  the  middle  term 
must  be  the  double  product  of  the  square  roots  of  the  two  others. 
Therefore,  to  obtain  the  square  root  of  a  trinomial  when  it  is  a  per- 
fect square ;  Extract  the  roots  of  the  two  exireme  terms,  and  give 
these  roots  the  same  or  contrary  signs,  according  as  the  middle  term 
is  positive  or  negative.  To  verify  it,  see  if  the  double  product  of  the 
two  roots  gives  the  middle  term  of  the  trinomial.  Thus, 
9a^— 48a*Z»2+64a^^*  is  a  perfect  square, 
suice  •v/9a^=3aS  and   V6^a^^=^—8aP, 

and  also     2  x  3a=  X  —  Sai^=  —  48a'*^'^     the  middle  term. 

But  4a-  +  14aZ'  +  9Z»^  is  not  a  perfect  square  :  for  although  4a^ 
and  +9^"^  are  the  squares  of  2a  and  3 J,  yet  2x2«X  3<5>  is  not  equal 
to  l^ab. 

3d.  In  the  series  of  operations  required  in  a  general  problem, 
when  the  first  term  of  one  of  the  remainders  is  not  exactly  divisi- 
ble  by  twice  the  first  term  of  the  root,  we  may  conclude  that  the 
proposed  polynomial  is  not  a  perfect  square.  This  is  an  evident 
consequence  of  the  course  of  reasoning,  by  which  we  have  arrived 
at  the  general  rule  for  extracting  the  square  root. 

4th.  When  the  polynomial  is  not  a  perfect  square,  it  may  be  sim- 
phfied  (See  Art.  ^25.). 

Take,  for  example,  the  expression      V  a^^+4rt^i^+4ai^. 

The  quantity  under  the  radical  is  not  a  perfect  square ;  but  it  can 
be  put  under  the  form  al{a^-\-4tab+4:P).  Now,  the  factor  between 
the  parenthesis  is  evidently  the  square  of  a-\-2b,  whence  we  may 
conclude  that, 

Va='&+4a^Z^+4ai*=  {a+2b)  Vab. 


RADICALS  OF  THE  SECOND  DEGREE.  141 

Of  the  Calculus  of  Radicals  of  the  Second  Degree. 

130.  A  radical  quantity  is  the  indicated  root  of  an  imperfect 
power. 

The  extraction  of  the  square  root  gives  rise  to  such  expressions 
as  va  ,  3  V  6  ,  7  V  2  ,  which  are  called  irrational  quantities,  or 
radicals  of  the  second  degree.  We  will  now  establish  rules  for  per- 
forming the  four  fundamental  operations  on  these  expressions. 

131.  Two  radicals  of  the  second  degree  are  similar,  when  the 
quantities  under  the  radical  sign  are  the  same  in  both.  Thus, 
zVh  and  5c  V~T  are  similar  radicals  ;  and  so  also  are  9  V  2 
and     7  VT. 

Addition  and  Subtraction. 

132.  In  order  to  add  or  subtract  similar  radicals,  add  or  subtract 
their  co-efficients,  then  prefix  the  siwi  or  difference  to  the  common 
radical. 

Thus,     .     .     .     3a  VT+5c  A/T=(3a+5c)  VT. 

And       .     .     .     3a  VT-5c  VT=(3a-5c)'/T. 

In  like  manner,       7  V2a+S  V2a=(l-{-S)  V2a—10  V2a. 

And       ...       7'/2^-3-v/2^=(7-3)  V^=   4^2^. 

Two  radicals,  which  do  not  appear  to  be  similar  at  first  sight, 
may  become  so  by  simplification  (Art.  125). 

For  example, 

V^8aP-{-l?  VT5^=U  V2a  +  5b  V3a=9b  Vs^, 

and     2 -/is  — 3  a/5^6 -/s"— 3  a/5^3  VsT 

When  the  radicals  are  not  similar,  the  addition  or  subtraction  can 
only  be  indicated.  Thus,  in  order  to  add  3  Vb  to  5  Va7 we  write 
5Va+2Vb7 


142 


Multiplication. 

133.  To  multiply  one  radical  by  another,  multiply  the  two  quan- 
tities under  the  radical  sign  together,  and  place  the  common  radical 
over  the  product. 

Thus,  Vax  Vh=  "v/o^;  this  is  the  principle  of  Art.  125,  taken  in 
an  inverse  order. 

When  there  are  co-efficients,  we  Jirst  multiply  them  together,  and 
write  the  product  before  the  radical.     Thus, 

3  Vbab'x^  V20a  =12  Vl00a''b'=120a  Vb7 

2a  V^  x3a  VTc=6a^  y/Vc"   =6a^c. 

2a  VaF+¥x—^a  Va-+^=  — 6a=(a^+^). 

Division. 

134.  To  divide  one  radical  by  another,  divide  one  of  the  quantu 
ties  under  the  radical  sign  hy  the  other  and  place  the  common  radical 
over  the  quotient. 

V  a 

vf 


Thus,      -  ^ —  N/  -j- ;     for  the  squares  of  these  two  expres- 

a 
sions  are  equal  to  the  same  quantity  —  ;    hence    the    expressions 

themselves  must  be  equal.     When  there  are  co. efficients,  write  their 
quotient  as  a  co-ejicient  of  the  radical. 

For  example, 

—   5a      /  h 


baVh^2hVc=~s/- 


2b 

\2ac  Vohc^^c  a/2^=3«  V-— -=3a  V^c. 
2o 

135.  There  arc  two  transformations  of  frequent  use  in  finding  the 
numerical  values  of  radicals. 


RADICALS  OF  THE  SECOND  DEGREE.  143 

The  first  consists  in  passing  the  co-efficient  of  a  radical  under  the 
sign.  Take,  for  example,  the  expression  3a  V  56  ;  it  is  equiva- 
lent  to  V  9a-  X  V^ib,  or  V  Qa?.bb  —  V  4:5a%  by  applymg 
the  rule  for  the  multiplication  of  two  radicals ;  therefore,  to  pass 
the  co-efficient  of  a  radical  under  the  sign,  it  is  only  necessary  to 
square  it. 

The  principal  use  of  this  transformation,  is  to  find  a  number 
which  shall  differ  from  the  proposed  radical,  by  a  quantity  less  than 
unity.  Take,  for  example,  the  expression  6  VlS;  as  13  is  not  a 
perfect  square,  we  can  only  obtain  an  approximate  value  for  its  root. 
This  root  is  equal  to  3,  plus  a  certain  fraction ;  this  being  multiplied 
by  6,  gives  18,  plus  the  product  of  the  fraction  by  6 ;  and  the  en- 
tire part  of  this  result,  obtained  in  this  way,  cannot  be  greater  than 
18.     The  only  method  of  obtaining  the  entire  part  exactly,  is  to  put 

6  Vl3  under  the  form Vg-xIS  =   VS6xl2=  V  468. 

Now  468  has  21  for  the  entire  part  of  its  square  root ;  hence,  6  VTs 
is  equal  to  21,  plus  a  fraction. 

In  the  same  way,  we  find  that  12  ^7=31,  plus  a  fraction. 

136.   The  object  of  the  second  transformation  is  to  convert  the 

«  «         .  . 

denominators  of  such  expressions  as  — ; — — ,  — ,  into  rational 

P+  Vq  V—  Vq 

quantities,  a  and  p  bemg  any  numbers  whatever,  and  q  not  a  per. 

feet  square.     Expressions  of  this  kind  are  often  met  with  in  the 

resolution  of  equations  of  the  second  degree. 

Now  this  object  is  accomplished  by  multiplying  the  two  terms  of 

the  fraction  by  p—  y/q,  when  the  denominator  is  ^+  y/q,  and  by 

i>+  \/?j  when  the  denominator  is  _p—  ^q.     For  multiplying  in  this 

manner,  and  recollecting  that  the  sum  of  two  quantities,  multiphed 

by  their  difference,  is  equal  to  the  difference  of  their  squares,  we 

have 

a o-iV—  \fq)  (i{p—  Vq)_ap—a  Vq 

P+Vq~{p+Vq){p-Vq)'~    f-q    ~    f-q  ' 


.» 


«»— ^»r.  *J^-«^    ^- 


--  — If —  I.  frbcsiHL     ^ 


If*- 


aoa  tae  -rstOK  ic 


,'5,11-,'^,     T^'S-T^'IS 


-.':— ^'? 


-i  Ifev  4i=  '•  ^3lS^r=^l^'L  »xiai  Mi, 


7^*5  sue— rrjLl      i4^ 


^4l 


RADICALS  OF  THE  SECOND  DEGREE.  145 

3+2  v/7 


:  2,123,  exact,  to  within  0,001. 


5x/12-6V5 

Remark.  Expressions  of  this  kind  might  be  calculated  by  ap- 
proximating to  the  value  of  each  of  the  radicals  which  enter  the 
numerator  and  denominator.  But  as  the  value  of  the  denominator 
would  not  be  exact,  we  could  not  form  a  precise  idea  of  the  degree 
of  approximation  which  would  be  obtained,  whereas  by  the  method 
just  indicated,  the  denominator  becomes  rational,  and  we  always 
know  to  what  degree  the  approximation  is  made. 

The  principles  for  the  extraction  of  the  square  root  of  particular 
numbers  and  of  algebraic  quantities,  being  established,  we  will  pro- 
ceed to  the  resolution  of  problems  of  the  second  degree. 

Examples  in  the  Calculus  of  Radicals. 

1.  Reduce    V  125    to  its  most  simple  terms. 

Ans.     5  V~b. 


v/^ 


2.  Reduce    v    — rr-  to  its  most  simple  terms. 
147  ^ 

5      /— 
Ans.     ^  v  6  . 

3.  Reduce    V  QSa^x    to  its  most  simple  terms. 

Ans,     la  V2Jc. 


4.  Reduce    V{x^—a-s(?)  to  its  most  simple  terms. 

5.  Required  the  sum  of   Vl^  and    V  128  . 

Ans.     14  V~2. 

6.  Required  the  sum  of   V^    and     V  147  . 

Ans.     10  VY 

.   /T  .  /27 

7.  Required  the  sum  of  \/  —    and    \^  — . 


50 


19    /_ 
Ans.     -VT. 


13 


8.  Required  the  sum  of  2  "v/  arb     and  3  V^^bx'^. 

9.  Required  the  sum  of  9  V  243    and  10  V  363. 

10.  Required  the  difference  of  \/  —  and  S/  ^. 

4      ,_ 
A71S.      T^  V  15. 
45 

11.  Required  the  product  of  5  V  8    and  3  V  5  . 

Arts.     30  VlO. 

2       /~1  3       /^ 

12.  Required  the  product  of  -^r-W    -^    and   —r\/  777. 

00  4  10 

Ans.     —  -/35. 
40 

13.  Divide  6  Vlo  by  3  V~b.  Ans.     2  VT. 

Of  Equations  of  the  Second  Degree. 

137.  When  the  enunciation  of  a  problem  leads  to  an  equation  of 
the  form  ax^=b,  in  which  the  unknown  quantity  is  multiplied  by 
itself,  the  equation  is  said  to  be  of  the  second  degree,  and  the  princi- 
ples established  in  the  two  preceding  chapters  are  not  sufficient  for 
the  resolution  of  it ;  but  since  by  dividing  the  two  members  by  a,  it 

b 
becomes  x^= — ,  we  see  that  the  question  is  reduced  to  finding  the 

b 
square  root  of   — . 
a 

138.  Equations  of  the  second  degree  are  of  two  kinds,  viz.  equa- 
tions  involving  two  terms,  or  incomplete  equations,  and  equations  in- 
volving three  terms,  or  co?iipIete  equations. 

The  first  are  those  which  contain  only  terms  involving  the  square 

of  the  unknown  quantity,  and  known  terms ;  such  are  the  equa- 

tions, 

1  5  7  299 

3,^=5;  -^-3  +  --.---.-+—. 


EQUATIONS  OF  THE  SECOND  DEGREE.  147 

These  are  called  equations  involving  two  terms  because  they  may- 
be reduced  to  the  form  aaP—b,  by  means  of  the  two  general  trans- 
formations (Art.  90  &  91).  For,  let  us  consider  the  second  equa- 
tion,  which  is  the  most  complicated  ;  by  clearing  the  fractions  it  be- 
comes 

8x2_72  +  10a^=7-24r'+299, 

or  transposing  and  reducing 

42x^=378. 
Equations  involving  three  terms,  or  complete  equations,  are  those 
which  contain  the  square,  and  also  the  first  power  of  the  unknown 
quantity,  together  with  a  known  term ;  such  are  the  equations 

5  13  2  ,      273 

5x2-7x=34  ;  y^- Y''»''+^=Q-y^'-^  +-12~' 

They  can  always  be  reduced  to  the  form  a3r+hx=c,  by  the  two 
transformations  already  cited. 

Of  Equations  involving  two  terms. 

139.  There   is   no  difficuhy  in  the  resolution  of  the  equation 

b 
ax?=h.     We  deduce  from  it  ar= — ,  whence  x-. 


'  ■      V^- 


When    —   is  a  particular  number,  either  entire  or  fractional, 

we  can  obtain  the  square  root  of  it  exactly,  or  by  approxmiation. 

If  —     is  algebraic,  we  apply  the  rules  established  for  algebraic 

quantities. 

But  as  the  square   of  +m   or    —m,   is    +m^,   it  follows  that 

|dtv/ — j    is  equal  to  — .     Therefore,  x  is  susceptible  of  two 

values,  viz.  a;=+Y/ — ,  and  x=  —  \/ — .      For,   substituting 
either  of  these  values  in  the  equation  aa^=  J,  it  becomes 


148  ALGEBRA. 

ax(  +  \/-)=h,ovax~=b, 

I  /*  \^  * 

and   .     .     .     flXf  — \/ — )  —h  or  ax — =b. 

For   another  example  take  the  equation  4a;^— 7=3a;^+9 ;  by- 
transposing,  it  becomes,  a;^=rl6,  whence  a;=±  vl6=±4. 
Again,  take  the  equation 

1  5  7  299 

We  have  already  seen  (Art.  138.),  that  this  equation  reduces  to 

378 
42a;^=378,  and  dividing  by  42,  x^r=— — =9;  hence  a;=±3. 

Lastly,  from  the  equation  Z3?=h  ;  we  find 

.=  ±V/-|-=±|\/15. 

As  15  is  not  a  perfect  square,  the  values  of  x  can  only  be  deter- 
mined by  approximation 

OJ  complete  Equations  of  the  Second  Degree. 
140.  In  order  to  resolve  the  general  equation 

we  begin  by  dividing  both  numbers  by  the  co-efficient  of  i^,  which 
gives, 

x^4- — X— — ,     or     o(f+x>x=q 
a         a 

b  ^    c 

by  makmg  — =p  and  — =?• 

Now,  if  we  could  make  the  first  member  aP+px  the  square  of  a 
binomial,  the  equation  might  be  reduced  to  one  of  the  first  degree, 
by  simply  extracting  the  square  root.  By  comparing  this  member 
with  the  square  of  the  binomial  (x+a),  that  is,  with  x^+Sax+a-, 
It  is  plam  that  x^+px  is  composed  of  the  square  of  a  first  term  x, 


EQUATIONS  OF  THE  SECOND  DEGREE.  149 

plus  the  double  product  of  this  first  term  x  by  a  second,  which  must 

P  P  r     P  P^        1. 

be  ^,  since  px=2^x',  therefore,  if  the  square  of  —  or  — ,  be 
added  to  x'^+px,  the  first  member  of  the  equation  will  become  the 
square  of  a;+^ ;  but  in  order  that  the  equality  may  not  be  destroy. 

ed     —     must  be  added  to  the  second  member. 
4 

By  this  transformation,  the  equation  x^+px=q  becomes 

p^  p^ 

Whence  by  extracting  the  square  root 


The  double  sign  ±  is  placed  here,  because  either 
+  V    ?+-7->  or  —V    $+"r'  squared  gives  5+—. 


p 

Transposing  — ,  we  obtain 


■I-±n/«+^' 


«  +  '4 

From  this  we  derive,  for  the  resolution  of  complete  equations  of 
the  second  degree,  the  following  general 

RULE. 

After  reducing  the  equation  to  the  form  x^-{-^x=<i,  add  the  square 
of  half  of  the  co-efficient  of  x,  or  of  the  second  term,  to  loth  man- 
bers ;  then  extract  the  square  root  of  both  members,  giving  the  double 
sign  db  to  the  second  member ;  then  find  the  value  of  x  from  the  re. 
suiting  equation. 

This  formula  for  the  value  of  x  may  be  thus  enunciated. 

The  value  of  the  unknown  quantity  is  equal  to  half  the  co-efficient 
13* 


150  ALGEBRA. 

of  X,  taken  with  a  contrary  sign,  plus  or  minus  the  square  root  of  the 
known  term  increased  hy  the  square  of  half  the  co-efficient  of  x. 
Take,  for  an  example,  the  equation 

5  13  2  273 

Clearing  the  fractions,  we  have 

10a;2-6a;+9=96-8x- 12x^+273, 
or,  transposing  and  reducing, 

22ar'+2x=360, 
and  dividing  both  members  by  22, 

2    _  360 
'''''+22'''"'22'- 

Add    (— ]    to  both  members,  and  the  equation  becomes 

2        /  1  x2      360      / 1  \2 
'^+22''+ (22/  ""^2~+l22/   ' 
whence,  by  extracting  the  square  root, 

1      ^  /360    Tr> 

^+22==^'^-22-  +  y' 
Therefore, 

^=-22^^-22-  +  (22)' 

which  agrees  ^vith  the  enunciation  given  above  for  the  double  value 

of  a;. 

It  remains  to  perform  the  numerical  operations.     In  the  first 

360       /  1  \^ 
place,     00   "^(oo)    "''ust  be  reduced  to  a  single  number,  having 

2*  \Zii 

(22)^  for  its  denominator. 

360       /I  v2_360x22  +  l_792I 
^°^^'  ^2~ ^  (22/  ~        (22)2       -  (2^  5 

extracting  the  square  root  of  7921,  we  find  it  to  be  89  ;  therefore, 


^       22   ^V22/       22 


EQUATIONS  OF  THE  SECOND  DEGREE.  151 

1    .  89 


Consequently, 

X- 

~~22     22' 

Separating  the  t\v 

■0  values, 

we  have 

1 
'"=-22- 

89     88 
"^22~22~' 

1 

"=-22 

89         45 
22~-ll' 

Therefore,  one  of  the  two  values  which  will  satisfy  the  proposed 
equaticJn,  is  a  positive  whole  number,  and  the  other  a  negative  frac- 
tion. 

For  another  example,  take  the  equation 

which  reduces  to 


ar. 

37 

57 
~~6' 

37 

/37s 

If  we  add  the  square  of   — ,  or   ( —  j     to  both  members,  it  be- 

comes 

37        /37\2  57      /37\2 

whence,  by  extracting  the  square  root 

37  ^  /     57     737^2 

Consequently, 

37^ .  /     57     /37x2 

In  order  to  reduce    l—j  — —    to  a  single  number,  wo  will  ob- 
serve, that 

(12)2=12x12=6x24; 
therefore,  it  is  only  necessary  to  multiply  57  by  24,  then  37  by  itself, 
and  divide  the  difference  of  the  two  products  by  (12)*.     Now, 
37x37=1369;  57x24=1368; 


152  ALGEBRA. 

therefore, 

/37\2     57_    1 
\12/  '~"6"~(12)''* 

1 

the  square  root  of  which  i 


12 


37      1 
Hence,      ^=j;^±j^,    or 


37      1  _38_19 
'l2"^12~12~'6"' 

37      1  _36_ 
'l2'~12~"l2~ 


This  example  is  remarkable,  as  both  of  the  values  are  positive, 
and  answer  directly  to  the  enunciation  of  the  question,  of  which 
the  proposed  equation  is  the  algebraic  translation. 

Let  us  now  take  the  literal  equation 

4a=^— 2a;2+2aa;=:18aJ— 18&2. 

By  transposing,  changing  the  signs,  and  dividuig  by  2,  it  becomes 

whence,  completing  the  square, 

ar'-ax+—=— Oah+dl/'. 

extracting  the  square  root, 


2  4 

9a2  3a 

Now,  the  square  root  of  — 9ab+9P,  is  evidently,  -^  — 3o. 

Therefore, 

a    ,  /3a         \  (  0.'=     2a  — db, 

x=—±{—-Sb),  or  ,  „, 

2      \  2         /  (  x=—  a+2b. 

These  two  values  will  be  positive  at  the  same  time,  if  2a>3i, 

and  dbya,  that  is  if  the  numerical  value  of  b  is  greater  than 

a  2a 

-77    and  less  than  — . 
Jo 


EQUATIONS  OF  THE  SECOND  DEGREE.  153 

EXAMPLES, 

x=2 


a:^— 7x+10=0  ....  values    . 

1  ^              4   ^                           i  x=     7,12  }  to  within 

3  '5                       ^       (  a;=-5,73  )       0,01.- 

3.  Given  a,-^— 8a;+10=19,  to  find  a-.                  Ans.     x=9.    ' 

4.  Given  a;^—a;— 40=170,  to  find  a;.                Ans.     a;r=15. 

5.  Given  Sa'^+Sa-— 9=76,  to  find  «.                 Ans.     x=5. 

6.  Given  ix^— i.r+7f=8,  to  find  a\                Ans.     a?=li. 

7.  Given    a^-{-P—2bx+x^=—^    to  find  x. 

n 

Ans.     x=-^—(b7iziz  Va'nv'+b^m^-a'lA. 
n-'—iir  \  I 

QUESTIONS. 

1.  Find  a  number  such,  that  twice  its  square,  increased  by  three 
times  this  number,  shall  give  65. 

Let  a:  be  the  unknown  number,  the  equation  of  the  problem  will  be 
2a^+3a^=65, 
whence. 


Therefore, 

3  23  3      23  13 

a;= — —+—=5,  and  x=— — =  — — . 

4  4  4       4  2 

Both  these  values  satisfy  the  question  in  its  algebraic 
For,  2x(5)='+3x5=2x25  +  15=:65. 

/     13x2  13      109      39      130 

and       2(--)  +3x-y=— -y-^~=65. 

But,  if  we  wish  to  restrict  the  enunciation  to  its  arithmetical 
sense,  we  will  first  observe,  that  when  k  is  replaced  by  —a?,  in  the 


154  ALGEBRA. 

equation  2x^+3a-'=65,  the  sign  of  the  second  term  3x  only,  is  chang- 
ed, because    {—xy=x^. 

3      23 
Therefore,  instead  of  obtaining  x= — T^X'     ^^   would    find 

3      2S  13 

x=—±—,  or  x=—-  and  a;=— 5,  values  which  only  differ  from 

the  preceding  by  their  signs.     Hence,  we  may  say  that  the  nega- 

13 

tive  solution    — — ,  considered  independently  of  its  sign,  satisfies 

this  new  enunciation,  viz.  :  To  find  a  number  such,  that  twice  its 
square,  diminished  hy  three  times  this  numler,  shall  give  65.  In  fact, 
we  have 

/13v2  13      169      39 

2.  A  certain  person  purchased  a  number  of  yards  of  cloth  for 
240  cents.  If  he  had  received  3  yards  less  of  the  same  cloth,  for 
the  same  sum,  it  would  have  cost  him  4  cents  more  per  yard.  How 
many  yards  did  he  purchase  ? 

Let  x=  the  number  of  yards  purchased. 

240 
Then will  express  the  price  per  yard. 

If,  for  240  cents,    he  had  received  3  yards  less,  that  is  x— 3 

yards,  the  price  per  yard,  in  this  hypothesis,  would  have  been  repre- 

240 
sented  by -.     But,  by  the  enunciation  this  last  cost  would  ex- 

X —  o 

ceed  the  first,  by  4  cents.     Therefore,  we  have  the  equation 
240       240  _ 

x-Z        ^"^^^ 

whence,  by  reducing  a,^— 3a;=180, 


3    .   ./''d        _       3±27 


x=— ±V  -r+i80=- 


2        '      4   '  2 

therefore 

a'=15,  and  x=  — 12. 


EQUATIONS  OP  THE  SECOND  DEGREE.  155 

The  value  a;=15  satisfies  the  enunciation  ;  for,  15  yards  for  240 

240 
cents,   gives        ,  ,  or  16  cents  for  the  price  of  one  yard,  and  12 

yards  for  240  cents,  gives  20  cents  for  the  price  of  one  yard, 
which  exceeds  16  by  4. 

As  to  the  second  solution,  we  can  form  a  new  enunciation,  with 
which  it  will  agree.  For,  go  back  to  the  equation,  and  change  x 
into  —X,  it  becomes, 

240         240  240       240  _ 

an  equation  which  may  be  considered  the  algebraic  translation  of 
this  problem,  viz. :  A  certain  person  'purchased  a  numler  of  yards 
of  cloth  for  240  cents  :  if  he  had  paid  the  same  sum  for  3  yards 
more,  it  would  have  cost  Mm  4  cents  less  per  yard.  How  many 
yards  did  he  purchase  ?  Ans.     «=12,  and  a:=  — 15. 

Remark.  Hence  the  principles  of  (Arts.  104  and  105.)  are 
confirmed  for  two  problems  of  the  second  degree,  as  they  were  for 
all  problems  of  the  first  degree. 

3.  A  merchant  discounted  two  notes,  one  of  $8776,  payable  in 
nine  months,  the  other  of  $7488,  payable  in  eight  months.  He 
paid  $1200  more  for  the  first  than  the  second.  At  what  rate  of 
interest  did  he  discount  them  ? 

To  simplify  the  operation,  denote  the  interest  of  $100  for  one 
month  by  x,  or  the  annual  interest  by  12x ;  9x  and  8a;  are  the  in- 
terests for  9  and  8  months.  Hence  100+9a;,  and  100+8a;,  repre- 
sent what  the  capital  of  $100  will  be  at  the  end  of  9  and  8  months. 
Therefore,  to  determine  the  present  values  of  the  notes  for  $8776, 
and  $7488,  make  the  two  proportions, 


100 +  9x  :  100  :  :  8776 

100 +  8x  :  100  :  :  7488 


877600 
100  +  9X 
748800 


100  +  8X  ' 
and  the  fourth  terms  of  these  proportions  will  express  what  the  mer- 


1 56  ALGEBRA. 

chant  paid  for  each  note.     Hence,  we  have  the  equation 
877600         748800 


1200; 


100  +  9a;        lOO+So; 

or,  observing  that  the  two  members  are  divisible  by  400, 
2194  1872 


100  +  9a;        100  +  8a; 
Clearing  the  fraction,  and  reducing,  it  becomes, 
216a'2+4396x==2200; 
whence 


2198/2200     (2198)2 

^        OTA      ' 


216        ^      216       (216)2 
Reducmg  the  two  terms  under  the  radical  to  the  same  denomi- 
nator. 


—  2198±  V  5306404 
ar— — 


216 

or  multiplying  by  12, 


-2198d=V  5306404 
12.= . 

To  obtain  the  value  of  12a;  to  within  0,01,  we  have  only  to  ex- 
tract  the  square  root  of  5306404  to  within  0,1,  since  it  is  afterwards 
to  be  divided  by  18. 

This  root  is  2303,5  ;  hence 

-2198±2303,5 


12a;= 

18 

and 

consequently. 

12a;= 

105,5 

18 

.-:.5,86, 

and 

12x— - 

-4501,5 

=  -250,08 

18 

The  positive  value,  12a;=5,86,  therefore  represents  the  rate  of 
interest  sought. 


EQUATIONS  OF  THE  SECOND  DEGREE.  157 

As  to  the  negative  solution,  it  can  only  be  regarded  as  connected 
with  the  first  by  an  equation  of  the  second  degree.  By  going  back 
to  the  equation,  and  changing  x  into  —x,  we  could  with  some  trou-^ 
ble,  translate  the  new  equation  into  an  enunciation  analogous  to  that 
of  the  proposed  problem. 

4.  A  man  bought  a  horse,  which  he  sold  after  some  time  for  24 
dollars.  At  this  sale,  he  loses  as  much  per  cent,  upon  the  price  of 
his  purchase,  as  the  horse  cost  him.  What  did  he  pay  for  the 
horse  ? 

Let  X  denote  the  number  of  dollars  that  he  paid  for  the 
horse,   a:— 24   will   express   the   loss   he   sustained.     But   as   he 

X 

lost  X  per  cent,  by  the  sale,  he  must  have  lost     -^^     upon  each 

ar 
dollar,  and  upon  x  dollars  he  loses  a  sum  denoted  by     -y-^- ;    we 

have  then  the  equation 

J?^=a;— 24,  whence  0,-2—1000;= -2400. 


and  a;=50±  V2500-2400=50±10. 

Therefore, 

=  60  and  a;=40. 

Both  of  these  values  satisfy  the  question. 

For,  in  the  first  place,  suppose  the  man  gave  $60  for  the  horse 

and  sold  him  for  24,  he  loses  36.     Again,  from  the.  enunciation,  he 

60  60x60 

should  lose  60  per  cent,  of  60,  that  is,     -— -     of  60,  or  ■  , 

which  reduces  to  36 ;  therefore  60  satisfies  the  enunciation. 

If  he  paid  $40  for  the  horse,  he  loses   16  by  the  sale  ;  for,  he 

40 
should  lose  40  per  cent,  of  40,  or  40  X  ,^..  >    which  reduces  to  16  ; 

therefore  40  verifies  the  enunciation. 

5.  A  grazier  bought  as  many  sheep  as  cost  him  £60,  and  after 

14 


158  ALGEBRA. 

reserving  fifteen  out  of  the  number,  he  sold  the  remamder  for  £54, 
and  gained  2s  a  head  on  those  he  sold  :  how  many  did  he  buy  ? 

Alls.     75. 

6.  A  merchant  bought  cloth  for  which  he  paid  £33  155,  which 
he  sold  again  at  £2  8s  per  piece,  and  gained  by  the  bargain  as 
much  as  one  piece  cost  him  :  how  many  pieces  did  he  buy  ? 

Ans.     15. 

7.  What  number  is  that,  which,  being  divided  by  the  product  of 
its  digits,  the  quotient  is  3 ;  and  if  18  be  added  to  it.  the  digits  will 
be  iirverted  ?  Ans.     24. 

8.  To  find  a  number  such  that  if  you  subtract  it  from  10,  and 
multiply  the  remainder  by  the  number  itself,  the  product  shall  be  21. 

Ans.     7  or  3. 

9.  Two  persons,  A  and  B,  departed  from  different  places  at  the 
same  time,  and  travelled  towards  each  other.  On  meeting,  it  ap- 
peared that  A  had  travelled  18  miles  more  than  B ;  and  that  A 
could  have  gone  B's  journey  in  15f  days,  but  B  would  have  been 
28  days  in  performing  A's  journey.     How  far  did  each  travel  ? 

>  -  jr  ^^^  ,  (  A  72  miles. 

^  (  B  54  miles. 


Discussion  of  the  General  Equation  of  the  Second  Degree. 

141.  As  yet  Ave  have  only  resolved  problems  of  the  second  de- 
gree, in  which  the  known  quantities  were  expressed  by  particular 
numbers.  To  be  able  to  resolve  general  problems,  and  interpret 
all  of  the  results  obtained,  by  attributing  particular  values  to  the 
given  quantities,  it  is  necessary  to  resume  the  general  equation  of 
the  second  degree,  and  to  examine  the  circumstances  which  result 
from  every  possible  hypothesis  made  upon  its  co-efficients.  This  is 
the  object  of  the  discussion  of  the  equation  of  the  second  degree. 

142.  A  root  of  an  equation  of  the  second  degree,  is  such  a  num. 
ber  as  being  substituted  for  the  unknown  quantity,  will  satisfy  the 
equation. 


EQUATIONS  OF  THE  SECOND  DEGIIEE.  159 

It  has  been  shown  (Art.  138),  that  every  equation  of  the  second 
degree  can  be  reduced  to  the  form 

x'+px^q  ....  (1), 
p  and  q  being  numerical  or  algebraic  quantities,  whole  numbers 
or  fractions,  and  their  signs  plus  or  minus. 
-  If,  in  order  to  render  the  first  member  a  perfect  square,  we  add 

—  to  both  members,  the  equation  becomes 
ir  jr 


(.+1)^=,+^ 


4* 

Whatever  may  be  the  value  of  the  number  expressed  by  q+~ri 
its  root  can  be  denoted  by  m,  and  the  equation  becomes 

(x+^)  =m\     or    {x+^)  -«r=0. 

But  as  the  first  member  of  this  equation  is  the  difference  between 
two  squares,  it  can  be  put  under  the  form 

{x+^-m).(x+^+ii^=Q  ;  .  .  .  (2). 

in  which  the  first  member  is  the  product  of  two  factors,  and  the 
second  is  0.  Now  we  can  render  the  product  equal  to  0,  and  con- 
sequently  satisfy  the  equation  (2),  in  two  different  ways  :  viz. 

P  'P 

By  supposing    x-{-— — m=0,    whence    x= — ^+'>n. 

% 

P  ,  P 

or  supposing      x+— 4-m=0,     whence    x=—— — m. 

At  * 


Or  substituting  for  m  its  value. 


160  ALGEBRA. 

Now,  either  of  these  values,  being  substituted  for  x  in  its  cor- 
responding  factor  of  equation  (2)  will  satisfy  that  equation ;  and  as 
equation  (1)  will  always  be  satisfied  when  the  derived  equation  (2) 
is  satisfied,  it  follows,  that  either  value  will  satisfy  equation  (1). 
Hence  we  conclude, 

1st.  That  every  equation  of  the  second  degree  has  two  roots,  and 
only  two. 

.2d.  That  every  equation  of  the  second  degree  may  he  decomposed 
into  two  binomial  factors  of  the  first  degree  with  respect  to  x,  having 
X  for  a  common  term,  and  the  two  roots,  taken  with  their  signs 
changed,  for  the  second  terms. 

For  example,  the  equation  o?+^x—2S  =  0  being  resolved  gives 
.T=4  and  a;=  —  7  ;  either  of  which  values  will  satisfy  the  equation. 
We  also  have 

(a;_4)  (x  +  7)   =.x^  +  3.r-28. 
143.  If  we  designate  the  two  roots  by  x'  and  x",  we  have 

.■=_f+^;7?    and    .■=-lL^V^, 

by  adding  the  roots  we  obtain 

^''+""=       2    —       -     4 
and  by  multiplying  them  together,  we  have 

4-('4)=-*- 

Hence,  1st.  The  algebraic  sum  of  the  two  roots  is  equal  to  the  co. 
efficient  of  the  second  term  of  the  equation,  taken  ivith  a  contrary 
sign.  2d.  The  product  of  the  two  roots  is  equal  to  the  second  mem- 
ber of  the  equation,  taken  also  with  a  contrary  sign. 


EQUATIONS  OF  THE  SECOND  DEGREE.  161 

Remake.  The  preceding  properties  suppose  that  the  equatiou 
has  been  reduced  to  the  form  c(r-{-px=q ;  that  is,  1st.  That  every 
term  of  the  equation  has  been  divided  by  the  co-efficient  of  x^. 
2d.  That  all  the  terms  involving  x  have  been  transposed  and  ar- 
ranged in  the  first  member,  and  x^  made  positive. 

144.  There  are  four  forms,  under  which  the  equation  of  the  se- 
cond degree  may  be  written. 

x^-\-px=     q  (1) 

a^—px  =     q  (2) 

s^+px  =  —q  (3) 

3?—px=  —  q  (4). 

In  which  we  suppose  p  and  q  to  be  positive. 
These  equations  being  resolved,  give, 


=-f-v/  .4 

(1) 

=+f-v  ,4 

(2) 

=-f-^-4 

(3) 

-+^±V- 


•?+T       (*)• 


In  order  that  the  value  of  x,  in  these  equations,  may  be  found, 
cither  exactly  or  approximatively,  it  is  necessary  that  the  quantity 
under  the  radical  sign  be  positive  (Art.  126). 

f         . 
Now,    —    being  necessarily  positive,  whatever  may  be  the  sign 

of  ^,  it  follows,  that  in  \he  first  and  second  forms  all  the  values  of 
x  will  be  real.     They  will  be  determined  exactly,  when  the  quan- 

tity     ?+-T-     is  a  perfect  square,  and  approximatively  when  it  is 

not  so. 

In  the  first  form,  ihe  first  value  of  x,  that  is,  the  one  arising  from 
14* 


162  ALGEBRA. 

taking  the  plus  value  of  the  radical,  is  always  positive  ;  for  the 


radical  S/  q+—,     being  numerically  greater  than    — ,  the  ex- 

pression     — —  iir\/   q+—     is  necessarily  of  the  same  sign  tis 

that  of  the  radical.  For  the  same  reason,  the  second  value  is  es- 
sentially  negative,  since  it  must  have  the  same  sign  as  that  with 
which  the  radical  is  affected :  but  each  root,  taken  with  its  proper 
sign,will  satisfy  the  equation.  The  positive  value  will,  in  general, 
alone  satisfy  the  problem  understood  in  its  arithmetical  sense  ;  the 
negative  value,  answering  to  a  similar  problem,  differing  from  the 
first  only  in  this ;  that  a  certain  quantity  which  is  regarded  as  ad- 
ditive in  the  one,  is  subtractive  in  the  other,  and  the  reverse. 

In  the  second  form,  the  first  value  of  x  is  also  positive,  and  the 
second  negative,  the  positive  value  being  the  greater. 

In  the  third  and  fourth  forms,  the  values  of  x  will  be  imaginary 
when 

5'>— ,     and  reaZ  when     2'<--t-- 


.    /  ^  p 

And  since     v    —  ?+-t-     is  less  than     — ,     it  follows  that  the 

real  values  of  x  will  both  be  negative  in  the  third  form,  and  both 
positive  in  the  fourth. 

145.  The  same  general  consequences  which  have  just  been  re- 
marked,  would  follow  from  the  two  properties  of  an  equation  of  the 
second  degree  demonstrated  in  (Art.  143).     The  properties  are  : 

TJie  algehraic  sum  of  the  roots  is  equal  to  the  co-efficient  of  the  se- 
cond  term,  taken  with  a  contrary  sign,  and  their  product  is  equal  to 
the  second  member,  taken  also  with  a  contrary  sign. 

For,  in  the  first  two  forms,  q  being  positive  in  the  second  mem- 
ber, it  follows  that  the  product  of  the  two  roots  is  negative  :  hence, 
they  have  contrary  signs.     But  in  the  third  and  fourth  forms  q  being 


EQUATIONS  OF  THE  SECOND  DEGREE.  163 

negative  in  the  second  member,  it  follows  that  the  product  of  the 
two  roots  will  be  positive  :  hence,  they  will  have  like  signs,  viz.  both 
negative  in  the  third  form,  where  p  is  positive,  and  both  positive  in 
the  fourth  form  where  p  is  negative. 

Moreover,  since  the  sum  of  the  roots  is  affected  with  a  sign  con- 
trary  to  that  of  the  co-efficient  p  ;  it  follows,  that,  the  negative  root 
will  be  the  greatest  in  the  first  form,  and  the  least  in  the  second. 

146.  We  will  now  show  that,  when  in  the  third  and  fourth  forms, 

p^ 
we  have     ?>"^»     the  conditions  of  the  question  will  be  incompa- 
tible with  each  other,  and  therefore,  the  values  of  x  ought  to  be 
imaginary. 

Before  showing  this  it  will  be  necessary  to  establish  a  proposition 
on  which  it  depends  :  viz. 

If  a  given  number  be  decomposed  into  two  parts  and  those  parts 
multiplied  together,  the  product  will  be  the  greatest  possible  when 
the  parts  are  equal. 

Let  p  be  the  number  to  be  decomposed,  and  d  the  difference  of 
the  parts.     Then 

p       d 

-^+— =         the  greater  part  (Art,  32). 

p       d 

and  — — — =         the  less  part. 


their  product  (Art.  46). 


rf      d^ 
and  ^_      =P, 

4       4 

Now  it  is  plain  that  P  will  increase  as  d  diminishes,  and  that  it 
will  be  the  greatest  possible  when  d=0  :  that  is, 

p      p     p^ 

-^X— =—     is  the  greatest  product. 

147.  Now,  since  in  the  equation 

a^ — px= — q 
p  is  the  sum  of  the  roots,  and  q  their  product,  it  follows  that  q  can 


164  ALGEBRA. 

never  be  greater  than  — .     The  conditions  of  the  equation  there- 

fore  fix  a  limit  to  the  vakie  of  q,  and  if  we  make  ?>^5  we  express 

by  the  equation  a  condition  which  cannot  be  fulfilled,  and,  this  con- 
tradiction  is  made  apparent  by  the  values  of  x  becoming  imaginary. 
Hence  we  may  conclude  that, 

The  value  of  the  unknown  quantity  loill  always  he  imaginary  when 
the  conditions  of  tlie  question  are  incompatible  with  each  other. 

Remark.  Since  the  roots  of  the  equation,  in  the  first  and  second 
forms,  have  contrary  signs,  the  condition  that  their  sum  shall  be 
equal  to  a  given  number  p,  does  not  fix  a  limit  to  their  product : 
hence,  in  those  two  forms  the  roots  are  never  imaginary. 

148.  We  will  conclude  this  discussion  by  the  following  remarks. 

1st.  If  in  the  third  and  fourth  forms,  we  suppose  q=-ri  the  ra- 
dical part  of  the  two  values  of  x  becomes  0,  and  both  of  these 

p 

values  reduce  to  x=  ——•.the  two  roots  are  then  said  to  le  equal. 

p^ 
In  fact,  by  substituting  -—  for   q   in  the    equation,    it    becomes 

P' 

x^-\-px—  —  —,  whence 

P^  I       P\^ 

a^+pa;+— =0,  or  \x+-^)  =0. 

In  this  case,  the  first  member  is  i\ie  product  of  two  equal  factors. 
Hence  we  may  also  say,  that  the  roots  of  the  equation  are  equal, 
since  in  this  case  the  two  factors  being  placed  equal  to  zero,  give 
the  same  value  for  x. 

2d.  If,  in  the  general  equation,  x^-^px=q,  we  suppose  q^O, 

P      P 
the  two  values  of  x   reduce  to   x= — ^+~,  or  x=0,  and  to 

2       2 

P       P 
x=-—-Y,  or  x=-p. 


EQUATIONS  OF  THE  SECOND  DEGREE.  165 

In  fact,  the  equation  is  then  of  the  form  !x^-{-px=0,  or  x{x+p)  =  0, 
which  can  be  satisfied  either  by  supposing  x=0,  or  x-\-p—0, 
whence  x=  —p  :  that  is,  one  of  the  roots  is  0,  and  the  other  the 
co-efficient  of  x  taken  with  a  contrary  sign. 

3d.  If  in  the  general  equation  oc^+px=q,  we  suppose  p=0, 
there  will  result  x"=q,  whence  x=±  y/q  ;  that  is,  in  this  case  tlie 
Lwo  values  of  x  are  equal,  and  have  contrary  signs,  real  in  the  first 
and  second  forms,  and  imaginary  in  the  third  and  fourth. 

The  equation  then  belongs  to  the  class  of  equations  involving  two 
terms,  treated  of  in  (Art.  139). 

4th.  Suppose  we  have  at  the  same  time  p=0,  ^'=0  ;  the  equa- 
tion reduces  to  x?  —  Q,  and  gives  two  values  of  x,  equal  to  0. 

149.  There  remains  a  singular  case  to  be  examined,  which  is  often 
met  with  in  the  resolution  of  problems  of  the  second  degree. 

To  discuss  it,  take  the  equation  ax^-\-hx=c.  This  equation 
gives 

'= 2-a • 

Suppose  now,  that  from  a  particular  hypothesis  made  upon  the 
given  quantities  of  the  question,  we  have  a=0  ;  the  expression  for 
X  becomes 

0 
-i±:b  I     """¥' 

-Q-'    ^^^'"^"     1       __2b 
"       0' 

The  second  value  is  presented  under  the  form  of  infinity,  and 
may  be  considered  as  an  answer  when  the  proposed  questions  will 
admit  of  answers  in  infinite  numbers. 

0 

As  to  the  first    — ,    we  must  endeavour  to  interpret  it. 

By  multiplying  the  numerator  and  denominator  of  the  2d  mem- 
ber  of  the  equation 

-b+  Vlr'  +  iac              -b-  V¥T^c 
'= 2-a ^^     2a 


166 

we  obtain 


h2—(^p^Aac)  — 4ac 


2a{-i—  Vl^'+^ac    ■2a{  —  b—  V¥+lac 
-2c 


Vb"-{-^ac 
c 


by  dividing  by  2a, 


—  by  making  a=0. 


Hence  we  see  that  the  indetermination  arises  from  a  common  fac- 
tor  in  the  numerator  and  denominator. 

If  we  had  at  the  same  time  a=0,  b—0,  c=0,  the  proposed 
equation  would  be  altogether  indeterminate. 

This  is  the  only  case  of  indetermination  that  the  equation  of  the 
second  degree  presents. 

We  are  now  going  to  apply  the  principles  of  this  general  discus- 
sion  to  a  problem  which  will  give  rise  to  most  of  the  circumstances 
which  are  commonly  met  with  in  problems  of  the  second  degree. 

Problem  of  the  Lights. 


C"  A  C      B        a 

150.  Find  upon  the  line  which  joins  two  lights,  A  and  B,  of  dif- 
ferent intensities,  the  point  which  is  equally  illuminated  ;  admitting 
the  following  principle  of  physics,  viz.  :  The  intensity  of  the  same 
light  at  two  different  distances,  is  in  the  inverse  ratio  of  the  squares 
of  these  distances. 

Let  the  distance  AB  between  the  two  lights  be  expressed  by  a ; 
the  intensity  of  the  light  A,  at  the  units  distance,  by  b ;  that  of  the 
light  B,  at  the  same  distance,  by  c.  Let  C  be  the  required  point, 
and  make  AC=ix,  whence  BC:=a—x. 

From  the  principle  of  physics,  the  intensity  of  A,  at  the  zmity 
of  distance,  being  b,  its  intensity  at  the  distances  2,  3,  4,  &:c.,  is 

b       b       b 
—5    — >    — ,  &c.,  hence  at  the  distance  x  it  will  be  expressed  by 


PROBLEM  OF  THE  LIGHTS.  167 

h 

-5-.     In  like  manner,  the  intensity  of  B  at  the  distance  a—x,  is 

c 

7 —  ;     but,  by  the  enunciation,  these  two  intensities  are  equal 

(a— a;)-  '  '     -^  ^1 

to  each  other,  therefore  we  have  the  equation 
I  c 


x^      (a—xy 
Whence,  by  developing  and  reducing, 

(h  —  c)xr  —  2aix=  —a^h. 
This  equation  gives 


ab        ^  /    d'W  ceh 

h-c        ^     {i-cf      b-c 
or  reducing, 

a(h±  Vhc) 
b—c 

This  expression  may  be  simplified  by  observing,  1st.  that  ~b±  Vbc 
can  be  put  under  the  form  y/b.  y/b±z  ^b.  v/c,  or  ■//>(  v/^±  ^c) ; 
2d.  that  b—c={^bf—{'^cy={y/b+^c).{^b—^c.)  There- 
fore,  by  first  considering  the  superior  sign  of  the  above  expression, 
we  have 

a^b{s/b+Vc)  cisfb 

""^  (  Vb+  v/c).(  Vb-  v/c)  "  Vb-  Vc  ' 
In  like  manner  we  obtain  for  the  second  value, 
a  Vb(  Vb—  v/f )        _      a^b 
'"'"'(  Vb+  n/c).(  Vb-  Vc)  ~  Vb+  Vc  ' 
Hence,  we  have 

aVb^  f  a  Vc 


1st  .  .  .  X-- 
2d  .  .  .  X-. 


Vb-\-  Vc^     I  from  which  j  '       Vb+  Vc' 

aVb  I  we  obtain  |  —aVc 


Vb-Vcj  [  Vb-Vc 

1st.  Suppose  that  Z>>c. 


168  ALGEBRA. 

The  first  value  of  x,  is  then  positive  and  less  than 

a,  because     — -, is  a  proper  fraction  ;  thus  this  value  gives 

for  the  required  point,  a  point  C,  situated  between  the  points  A  and 
B.  We  see  moreover,  that  the  point  is  nearer  to  B  than  A  ;  for 
since    J>c,  we   have     s/h+ ^h   or    2  v/3>(  VJ+ v/c) ;  whence 

->—     and  consequently,  ^"o"' 


ought  to  be  the  case,  since  the  intensity  of  A  was  supposed  to  be 
greater  than  that  of  B. 

The  corresponding  value  of  a—x,         ,  is  also  positive, 

a 
and  less  than     — ,     as  may  easily  be  shown. 

a\/h        .       , 

The  second  value  of  x,  ——, -,  is  also  positive,  but  greater 

■\/o —  vc 

than  a  :  because  — -, r>l-     Hence  this  second  value  gives  a 

y/O—   \/C 

second  point  C,  situated  upon  the  prolongation  of  AB,  and  to  the 
right  of  the  two  lights.  We  may  in  fact  conceive  that  the  I'wo  lights, 
exerting  their  influence  in  every  direction,  should  have  upon  the 
prolongation  of  AB,  another  point  equally  illuminated ;  but  this 
point  must  be  nearest  that  light  whose  intensity  is  the  least. 

We  can  easily  explain,  why  these  two  values  are  connected  by 
the  same  equation.  If,  instead  of  taking  AC  for  the  unknown  quan- 
tity X,  we  had  taken  AC,  there  would  have  resulted  BC'=x—a  ; 

b  c 

and  the  equation  -^^^-r-^:^-     Now,  as  {x  —  ay  is  identical  with 

{a—xy,  the  new  equation  is  the  same  as  that  already  established, 
which  consequently  should  have  given  AC  as  well  as  AC. 

And  since  every  equation  is  but  the  algebraic  enunciation  of  a 
problem,  it  follows  that,  when  Ihe  same  equation  enunciates  several 
problems,  it  ought  by  its  different  roots  to  solve  them  all. 


EQUATIONS  OF  THE  SECOND  DEGREE.  169 

When  the  unknown  quantity   x    represents  the  liiie  AC,   the 
second  value  of  a  —  x,  — — ,  is  negative,  as  it  should  be,  since 

y/l) —   y/C 

we    have    a;>a ;     but    by    changing   the    signs    in   the    equation 

—  ay/c  as/c 

It  becomes  a;— a= — ~ —  ;  and  this  value  of 


x—a  represents  the  positive  value  of  BC . 

2d.     Let  3<c. 

a  s/h 
The  first  value  of  x,    — -r- — —    is  always  positive,  but  less  than 
■v/o+  vc 
a 
— ,     since  we  have 

(  v/i+  v/c)>(  v/Z»+  ^l)  or  than  2  ^l. 
The  corresponding  value  of  a—x,  or  — —  is  positive,  and 

y/0-\-  y/c 

greater  than  — . 

Therefore  in  this  hypothesis,  the  point  C,  situated  between  A 
and  B,  must  be  nearer  A  than  B. 

rri  1       1        r  "-^^  —as/l)       . 

i  he  second  value  of  x,  — —  or  — —,  is  essentially  ne- 

gative.  To  interpret  it,  let  us  take  for  the  unknown  quantity  the 
distance  AC",  and  let  us  represent  this  distance  by  a;,  and  at  the 
same  time  consider,  as  we  have  a  right  to  do,  x  as  essentially  ne- 
gative. Then  the  general  expression  for  BC"  being  a  —  x,  if 
we  regard  x  as  essenticdly  negative,  the  true  numerical  value  of 
a—x  is  expressed  by  a+x.  Hence  as  before,  the  equation  or 
algebraic  expression  will  be 


be  b 


or 


x^      {a-xf  a-2      (a  +  xf 

in  the  first  of  which  equations  x  is  essentially  negative. 

This  equation  ought  to  give  a  negative  value  for  x,  and  a  posi- 
tive  value  for  BC"=a-\-x.     Indeeed,  since  the  intensity  of  the  light 
B  is  greater  than  that  of  A,  the  second  required  point  ought  to  be 
15 


170  ALGEBRA. 

nearer   A   than   B.     The   algebraic    value   for   BC",   which  is 
—  a  Vc  a  Vc 


■\/h   —  Vc~  Vc    —  Vb~ 

3d.     Let  h=c. 


positive. 


a 
The  first  two  values  of  a;  and  a—x  reduce  to  — ,   which  gives 

the  middle  of  AB  for  the  first  required  point.  This  result  agrees 
with  the  hypothesis. 

The  two  other  values  reduce  to  — - — ,  or  infinity ;  that  is,  the 

second  required  point  is  situated  at  a  distance  from  the  two  points 
A  and  B,  greater  than  any  assignable  quantity.  This  result 
agrees  perfectly  with  the  present  hypothesis,  because,  by  supposing 
the  difference  b—c  to  be  extremely  small,  without  being  absolutely 
nothing,  the  second  point  must  be  at  a  very  great  distance  from  the 

a^/b 
lights  ;  this  is  indicated  by  the  expression  — —,  the  denomi- 
nator of  which  is  extremely  small  with  respect  to  the  numerator. 
And  if  we  finally  suppose  Z»=c,  or  y/b— ^/c=0,  the  required  point 
cannot  exist  for  a  finite  distance,  or  is  situated  at  an  infinite  distance. 
We  will  observe,  that  in  the  case  of  b=c,  if  we  should  consider 
the  values  before  they  were  simplified,  viz. 

a(b+-ybc)  a(b— y/bc) 


x=- 


-  ,  and  X—        J 

b~c  b—c 


aVb 
the    first,   which    corresponds    to     a;— — -, — ,    would  become 

2ab  a  s/b 

— r-,  and  the  second,  which  corresponds  to  — ; — ,    would   be- 

0  0 

come  — .  But  —  would  be  obtained  in  consequence  of  the  exist- 
ence of  a  common  factor,  yjb—  -/c,  between  the  two  terms  of  the 
value  of  X  (see  Art.  113). 


EQUATIONS  OF  THE  SECOND  DEGREE.  171 

Let  l=:c,  and  a  =  0. 
The  first  system  of  values  for  x  and  a—x,  reduces  to  0,  and  the 

second  to     — .     This  last  symbol  is  that  of  indeiermination ;  for, 

resuming  the  equation  of  the  problem,  {b—c)xP—'iabx=—a?l,  it 
reduces,  in  the  present  hypothesis  to  O.or  — 0.a;=0,  which  maybe 
satisfied  by  giving  x  any  value  whatever.  In  fact,  since  the  two 
lights  have  the  same  intensity,  and  are  placed  at  the  same  point, 
they  ought  to  illuminate  equally  each  point  of  the  line  A  B. 

The  solution  0,  given  by  the  first  system,  is  one  of  those  solutions 
in  infinite  numbers,  of  which  we  have  spoken. 

Finally,  suppose  a=0,  and  h  and  c,  unequal 

Each  of  the  two  systems  reduces  to  0,  which  proves  that  there  is 
but  one  point  in  this  case  equally  illuminated,  and  that  is  the  point 
in  which  the  two  lights  are  placed. 

In  this  case,  the  equation  reduces  to  {h—c)s^—0,  and  gives  the 
two  equal  values,  a;=0,  a;=0. 

The  preceding  discussion  presents  another  example  of  the  pre- 
cision  with  which  algebra  responds  to  all  the  circumstances  of  the 
enunciation  of  a  problem. 

Of  Equations  of  the  Second  Degree,  involving  two  or  more 
unknown  quantities, 

151.  A  complete  theory  of  this  subject  cannot  be  given  here,  be- 
cause the  resolution  of  two  equations  of  the  second  degree  involv- 
ing  two  unknown  quantities,  in  general  depends  upon  the  solution  of 
an  equation  of  the  fourth  degree  involving  one  unknown  quantity ; 
but  we  will  propose  some  questions,  which  depend  only  upon  the 
solution  of  an  equation  of  the  second  degi-ee  involving  one  unknown 
quantity. 

1.  Find  two  numbers  such  that  the  sum  of  their  products  by  the 
respective  numbers  a  and  l,  may  be  equal  to  2^,  ansl  that  their 
product  may  be  equal  to  p. 


172 


ALGEBRA. 


Let  X  and  y  be  the  required  numbers,  we  have  the  equations, 

ax-\-hy^2s. 

xy=p. 

2s— ax 
h  rom  the  first     y— — -- — ;     whence,  by  substituting  in  the  se- 
cond, and  reducing, 

aa^—2sx=z—hp. 
Therefore, 

-  a'= 
and  consequently. 


1     , 

—  V  s^  —  abp, 
a  ^ 


This  problem  is  susceptible  of  two  direct  solutions,  because 


S   IS 


evidently  >  Vs^~abj),  but  in  order  that  they  may  be  real,  it  is 
necessary  that  ^>  or  ^abp. 

Let  a=5=l ;  the  values  of  x,  and  y,  reduce  to 
x=s±:  Vs^—p     and    y—szp  Vs^—p 

Whence  we  see,  that  the  two  values  of  x  are  equal  to  those  of  y, 
taken  in  an  inverse  order  ;  which  shows,  that  if  «+  Vs^—p  repre- 
sents  the  value  of  a;,  s—  Vs^—p  will  represent  the  corresponding 
value  of  y,  and  reciprocally. 

This  circumstance  is  accounted  for,  by  observing,  that  in  this  par- 

ticular   case  the   equations  reduce  to      <  '     and  then  the 

question  is  reduced  to,  finding  two  numbers  of  which  the  sum  is  2s, 
and  their  product  p,  or  in  other  words,  to  divide  a  number  2s,  into 
two  such  parts,  that  their  product  may  be  equal  to  a  given  number  p. 

2.  Find  four  numbers  in  proportion,  knowing  the  sum  2s  of  their 
extremes,  the  sum  2s'  of  the  means,  and  the  sum  4c^  of  their  squares. 

Let  u,  X,  y,  z,  denote  the  four  terms  of  the  proportion  ;  the  cqua- 
tions  of  the  problem  will  be 


EQUATIONS  OF  THE  SECOND  DEGREE.  173 

u+z=2s 

x+y=2s' 

uz=xy 

At  first  sight,  it  may  appear  difficult  to  find  the  values  of  the  un- 
known quantities,  but  with  the  aid  of  an  unknown  auxiliary  they  are 
easily  determined. 

Let  p  be  the  unknown  product  of  the  extremes  or  means,  we 
have 

1st.  The  equations 

c  u+z=2s,       ,  .  ,       .  (  «=*+  V  s^-p, 

\  which  give  ,    \  , 

I      uz=p,  °  i  z=s-V s'-p. 

2d.  The  equations 

ix+y=2s',  .  tx=s'+Vs'^-p, 

\  which  give      \  , 

\       ^=P,  iy=s'-Vs'^-p. 

Hence,  we  see  that  the  determination  of  the  four  unknown  quan- 
tities depends  only  upon  that  of  the  product  p. 

Now,  by  substituting  these  values  of  u,  x,  y,  z  in  the  last  of  the 
equations  of  the  problem,  it  becomes 

+  (s'-  V7^^py=^c' ; 
or,  developing  and  reducing, 

452^45'2_4^_4c .     hence    p=s^+s"^—c''. 

Substituting  this  value  for  p,  in  the  expressions  for  u,  x,  y,  z,  we 
find 


,  u=s-\-  V  c'-s"',  (  x=s'-\-  V  c'-s', 

\  z=s-  V  c^-T^,  \  y=s'—  V c^-s^- 

These  four  numbers  evidently  form  a  proportion  ;  for  we  have 


15* 


174 


ALGEBRA. 


Tliis  problem  sliows  how  much  the  mtroduction  of  an  unknown 
auxiliary  facilitates  the  determination  of  the  principal  unknown  quan- 
tities. There  are  other  problems  of  the  same  kind,  which  lead  to 
equations  of  a  degree  superior  to  the  second,  and  yet  they  may  be 
resolved  by  the  aid  of  equations  of  the  first  and  second  degrees,  by 
introducing  unknown  auxiliaries. 

152.  We  will  now  consider  the  case  in  which  a  problem  leads  to 
two  equations  of  the  second  degree,  involving  two  unknown  quan- 
tities. 

An  equation  involving  two  unknown  quantities  is  said  to  be  of  the 
second  degree,  when  it  contains  a  term  in  which  the  sum  of  the  expo- 
nents of  the  two  unknown  quantities  is  equal  to  2.     Thus, 

S3r'—Ax+f—xy—5y+Q=0,     7xt/— 4x+3/=0, 
are  equations  of  the  second  degree. 

Hence,  every  general  equation  of  the  second  degree,  involving 
two  unknown  quantities,  is  of  the  form 

ay"  +  bxy + cx^ + dy  +fx  +g=0, 

a,  h,  c,  .  .  .  representing  known  quantities,  either  numerical  or  al- 
gebraic. 

Take  the  two  equations 

af+bxy  +  cx''-\-dy+fx+g=0, 
aY+i'xy-{-c'a^-\-d'y-\-f'x-\-g'=0. 
Arranging  them  with  reference  to  x,  they  become 

c  x/'  +  {by+f  )x+af+dy+g  =0, 
c'oir'  +  {h'y+f')x-{-ay-{-d'y+g'z=0. 
Now,  if  the  co-cfficients  of  x^  in  the  two  equations  were  the  same, 
wc  could,  by  subtracting  one  equation  from  the  other,  obtain  an 
equation  of  the  first  degree  in  x,  which  could  be  substituted  for  one 
of  the  proposed  equations  ;  from  this  equation,  the  value  of  x  could 
be  found  in  terms  of  y,  and  by  substituting  this  value  in  one  of  the 
proposed  equations,  we  would  obtain  an  equation  involving  only  the 
unknown  quantity  y. 


EQUATIONS  OF  THE  SECOND  DEGREE.  175 

By  multiplying  the  first  equation  by  c',  and  the  second  by  c,  they 
become 

ccV+(5?/+/)c'a;+(a/+cZy+g)c'=0, 
cc'a^  +  {h'y-{-f')c  x+(ay +d't/+^')c  =0, 
and  these  equations,  in  which  the  co-efficients  of  x^  are  the  same, 
may  take  the  place  of  the  preceding. 

Subtracting  one  from  the  othei',  we  have 
\{hc'  —  cb')y-\-fc'  —  cf''\x-\-{ac'  —  ca')'f  -\-{dc'  —  cd')y-\-gc'  —  eg' =0, 
which  gives 

{ca'  —  ac')y^-\-{cd'—dc')y-\-cg'—gc' 
~  {hc'—cb')y+fc'-cf' 

This  expression  for  x,  substituted  in  one  of  the  proposed  equa- 
tions, will  give  a  final  equation,  involving  y. 

But  without  effecting  this  substitution,  which  would  lead  to  a  very 
complicated  result,  it  is  easy  to  perceive  that  the  equation  involving 
y  will  be  of  the  fourth  degree  ;  for  the  numerator  of  the  expres- 
sion for  X  being  of  the  form  my^-\-ny-\-p,  its  square,  or  the  expres- 
sion for  x^,  is  of  the  fourth  degree.  Now  this  square  forms  one  of 
the  parts  of  the  result  of  the  substitution. 

Therefore,  in  general,  the  resolution  of  two  equations  of  the  se- 
cond degree,  involving  two  unknown  quantities,  depends  upon  that  of 
an  equation  of  the  fourth  degree,  involving  one  unknown  quantity. 

153.  There  is  a  class  of  equations  of  the  fourth  degree,  that  can 
be  resolved  in  the  same  way  as  equations  of  the  second  degree  ; 
these  are  equations  of  the  form  x'^-\-poc^-\-q=.0.  They  are  called 
trinomial  equations,  because  they  contain  but  three  kinds  of  terms  ; 
viz.  terms  involving  a;*,  those  involving  x^,  and  terms  entirely  known. 

In  order  to  resolve  the  equation  x'^-\-px^-{-q=Q,  suppose  a^=y» 
we  have 


Pa./  V^ 

f^Vy^i=^^  whence  3/=-y±V   —^.^-^^ 

But  the  equation  ar^=y,  gives  a;=±  s/y. 


176 


Hence,  x=±V  -^±\/_^+^. 

We  perceive  that  the  unknown  quantity  has  four  values,  since 
each  of  the  signs  +  and  — ,  which  affect  the  first  radical,  can  be 
combined  successively  with  each  of  the  signs  which  affect  the  se- 
cond  ;  but  these  values  taken  two  and  two  are  equal,  and  have  contra, 
ry  signs. 

Take  for  example  the  equation  a;*  — 25a;^=:  — 144  ; 

by  supposing  s^—y,  it  becomes  ?/^— 25^/=  — 144  ; 

whence  ^=16,  y^Q. 

Substituting  these  values  in  the  equation  x^=y  there  will  result 

1st.  3?=IQ,  whence  a;=:±4;     2d.  a^—g^  whence  a;=±3. 

Therefore  the  four  values  are  +4,  —4,   +3  and  —3. 

Again,  take  the  equation  a;^— 7x^=8.  Supposing  s?=y,  the 
equation  becomes  /— 7^=8;  whence  y=Q,  y=  —  l. 

Therefore,    1st.     !c^=8,   whence    a;=±2^/2;      2d.    ar^=-l; 
whence  x=  ±  v/— 1 ;  the  two  last  values  of  x  are  imaginary. 
Let  there  be  the  algebraic  equation     x'^—(2ic-\-4a'')x^=—lr'c'; 
taking  x^=y,  the  equation  becomes        f—{2ie-\-4:a^)y=—i^c''; 
from  which  we  deduce  y=         U  +  2a2±2a  VlT^^T^, 


And  consequently  x=±\y  be  ^2a'^±2a^/lc  +  a^. 

154.  Every  equation  of  the  form  y^''+py''+q=0,  in  which  the 
exponent  of  the  unknown  quantity  in  one  term  is  double  that  of  the 
other,  may  be  solved  by  the  rules  for  equations  of  the  second  degree. 

For,  put    y"=x,  then  f"=3?,  and  y^" -{-fy^ ■\-q=x'^ ^'px-{-q=^. 

Hence 
Or 

And 


a;  =    - 

-^4. 

r  =  - 

--!• 

=  V    - 

■i-^- 

^4- 

EQUATIONS  OF  THE  SECOND  DEGREE.  177 

Extraction  of  the  Square  Root  of  Binomials  of  the  form 
a  ±  VT^ 

155.  The  resolution  of  trinomial  equations  of  the  fourth  degree, 
gives  rise  to  a  new  species  of  algebraic  operation  :   viz.  the  extrac- 

tion  of  the  square  root  of  a  quantity  of  the  form  ad=  Vb,    a  and  h 
being  numerical  or  algebraic  quantities. 

By  squaring  the  expression  3±  Vs,    we  have 

(3zfc  Vy)2=9±6  VT+5=14±6  Vb~: 

hence,  reciprocally  \/  14±6  V  5  =3±  V  5. 

In  like  manner,  (  v/7±  ^/ll)2=7±2^/7x  \/ll  +  ll 
=  18±2^/77. 

Hence  reciprocally     V18±2  V77=  n/7±  v/11. 

Whence  we  see  that  an  expression  of  the  form  v  a=fc  \/h,  may 
sometimes  be  reduced  to  the  form  a'±  y/h'  or  -/a'db  V^' ;  and 
when  this  transformation  is  possible,  it  is  advantageous  to  effect  it, 
since  in  this  case  we  have  only  to  extract  two  simple  square  roots, 
whereas  the  expression  Vai  \/b  requires  the  extraction  of  the 
square  root  of  the  square  root. 

156.  If  we  let  p  and  q  denote  two  indeterminate  quantities,  we 
can  always  attribute  to  them  such  values  as  to  satisfy  the  equations 

Va+Vh:=p+q (1). 

Va—  y/b=p—q (2). 

These  equations,  being  multiplied  together,  give 

Va'-b^p^-q' (3). 

Now,  if  p  and  q  are  irrational  monomials  involving  only  single  ra- 
dicals  of  thesecond  degree, orif  one  is  rational  and  the  other  irration- 
al,  it  follows  that  p'^  and  q^  will  be  rational ;  in  which  case,  p^ — q", 

or  its  value,  Va'—b,  is  necessarily  a  rational  quantity,  or  a?—h  is 
a  perfect  square. 


178  ALGEBRA. 

When  this  is  the  case,  the  transformation  can  always  be  effected. 
For,  take  o?—b,  a  perfect  square,  and  suppose  Va^—l=c',  the 
equation  (3)  becomes 

Moreover,  the  equations  (1)  and  (2)  being  squared,  give 
f+q^+2fq=a+  ^/&, 
p^J^q^  —  2pq=a—^/h^, 
whence,  by  adding  member  to  member, 

f+f=^ (4) ; 

but  y^—f=c (5). 

Hence,  by  adding  these  last  equations,  and  subtracting  the  se- 
cond from  the  first,  we  obtain 

25'2=a-c; 


and  consequently 


p=±' 


q^±.' 


2    '■ 


2    ' 

~2 

f  +  C 


or 


Therefore, 

Va-\-Vb,     or    p+g-^riV   — ^db' 

V  a— \/b,     or     p—q—±:\/  — 

^"+v*=±(V-i-+V— ) 

/ /*    /o+c      .    /a  —  c\ 


a  —  c 
~2~' 

'a—c 
2     '■ 

.  .  (6\ 
.  .  (7). 


These  two  formulas  can  be  verified ;  for  by  squaring  both  mem- 
bers of  the  first,  it  becomes 

a-\-c      a  —  c        .    /  a^—c^  .— — - 

a+  v/i--^+-^-  +  2V  — ^— =:a+  Va'-c'  ; 

but  the  relation    Va^—h  —  c,  gives  c^z=a^—b. 


EQUATIONS  OF  THE  SECOND  DEGREE.  179 

Hence,  a+  Vi=a+  Va''—a^+b=a-{-  -Jh. 

The  second  formula  can  be  verified  in  the  same  manner. 

157.  Remark.  As  the  accuracy  of  the  formulas  (6)  and  (7)  is 

proved,  whatever  may  be  the  quantity  c,  or  Va^—h,  it  follows, 
that  when  this  quantity  is  not  a  perfect  square,  we  may  still  replace 

the  expressions  Va+  y/b  and  Va—  \/h,  by  the  second  inembers 
of  the  equalities  (6)  and  (7) ;  but  then  we  would  not  simplify  the 
expression,  since  the  quantities  p  and  q  would  be  of  the  same  form 
as  the  proposed  expression. 

We  would  not,  therefore,  in  general,  use  this  transformation, 
unless  c?—!)  is  a  perfect  square. 

EXAMPLES 

158.  Take  the  numerical  expression  94+42  v'5,  which  reduces 
to  94+  V8820.     We  have 

a=94,  5=8820, 
whence  c=  ■\/'cf—b=  V'8836  — 8820=4, 

a  rational  quantity  ;  therefore  the  formula  (6)  is  applicable  to  this 


It  becomes 

/ /.    /94+4 

•+' 


'/94+42  v/5=  ±  (  V   ^^^ 

01-,  reducing,  =^(  V^+  ^45)  ; 

therefore,  ■v/94T42V5=  ±(7+3  v/5). 

In  fact,         (7  +  3  v/5)-=49  +  45  +  42  v/5  =  94+42  v/5. 

Again,  take  the  expression 

S/  np+2m^—2m  Vnp+m^; 
we  have  a=np-\-2)H^,     b—'lm^{np-\-m^), 

whence  a^—i=7i^p^, 

and  c  or    Va'  — 3=7ip; 


180  ALGEBRA. 

therefore  the  formula  (7)  is  applicable.     It  gives  for  the  required 
root 


or,  reducing,  ±(  V  np-\-7n^—m). 

In  fact,        (  Vnp  +  7n^—my=np+2m^—2}n  V  np+n?. 
For  another  example,  take  the  expression 

V  16  +  30  V^T+Vie-so  V~^, 
and   reduce  it  to  its  simplest  terms.     By  applying  the  preceding 
formulas,  we  find 


V  16  +  30  a/-1  =  5  +  3V-1,     1/16-30^-1  =  5-3^31^ 

Hence,      V  16  +  30 '/^+ \/  16-30 -v/^  =  10. 

This  last  example  shows,  better  than  any  of  the  others,  the  utili- 
ty  of  the  general  problem ;  because  it  proves  that  imaginary  ex- 
pressions  combined  together,  may  produce  real,  and  even  rational 
results. 


\/28  +  10  VT=5+ VT;      V  1+4  V-3=:2+  V -S, 

\/  bc  +  2b  V  bc-W  +  \/  hc-2b  V  hc-lF^^2b; 

\/ ab  +  ^c^-d?+2  Viabc^-ab(P=  Vab+  VAit-'^. 

Examples  of  Equations  of  the  Second  Degree,  which  either 
involve    Radicals,  or  tivo  unknown  quantities. 

2a^ 

1.   Given     x-\-  V a~+3r  — — ==     to  find  x. 
Va^-\-y? 


X  Vc^x^+a'+x'=2d' 

X  'y/a^-\-a?=a^—3?  by  transposing. 


EQUATIONS  OF  THE  SECOND"  DEGREE. 


181 


henco 
or 


a!'a^+x*=a*  —  2a^x^+x\     by  squaring, 
2a\^=xK 


2.  Given 


x=dz' 
\/ ^^j^b'^-S/ ^-¥=h     to  find  X. 


J+^=v/^_^+,, 


by  transposing. 


^+^=^-^+2* 


.h'  +  h\ 


hence 


hence 


V=2b\/  -^-y. 


ar'=- 


4a^ 


hence a;=±- 


2a 


J  VT 


3.   Given 1 =-r     to  find  a?. 

a,'  X  0 


Ans.     a;=3±  V2ab  —  b^. 


4.  Given 


««/ 


VT 


and 


:48 


:24 


>       to  find  X  and  ?/. 


Dividing  the  first  equation  by  the  second,  we  have 
IG 


182 


V- 


V  y  =2,  or  t/=4. 


4a; 
Whence  from  the  second  equation =4  V~x=24, 


to  fino  X. 


G.  Given     x  +  -X^-^-y  =   19  ) 

and         x^+     xy+f  ^13S\       to  find  a:  and  y. 

Dividbg  the  second  equation  by  the  first,  we  have 
X—  ■Vxy-\-y=   7 
but        ...      x-\-  Vxy+y=l9 

hence  ....         2a:+2!/=26     by  addition, 
or         ....  a;+  ?/=13 

and       ....     Vxy-\-13  =  19     by  substituting  in  the  1st  eq. 

or         ....     ■Vxy=  6 

and       ....         xy=:26 

From  2d  equation,       x^+xy-{-y'^=123 

and  from  the  last  Sxy        =108 

Subtracting     .      .      oi^  —  2xy+y^—   25 

Hence x—y=zt:     5 

But x-\-y=       13 

Hence        .     .    x=9  or  4;     and     «/=4  or  9. 


7.   Gi 


a—  V  cr 


1+  -/a^-x^ 


:i,     to  find  X. 


Ans.     x=±- 


2a  VT 


EQUATIONS  OF  THE  SECOND  DEGREE.  183 

8.  Given         =- to  find  x. 

V  X  —  V x—a     *'~^ 

a(l±nY 
Ans.     — 


l±2w 


^.  V  a-\-x      V  a  —  x      .    /~x 

9.  Given     — ^H ;z^  =  V   ^     to  find  x. 

V  X  V  X  ^ 


Ans.     x=±2Vab-¥. 

10.  Given   j  ^,3^^^^^i2^  |    to  find  x  and  y. 

(  a;z=2     or     1 
Ans.      { 

'  y=l     or     2. 

11.  Given    j  ,  „      2, ,        -      _„_         to  find  a;  and  y. 

(  x=ll     or     5 

Ans.      \        ^  ,, 

(  y=b     or     11. 


a+a;+  V2aa;+x2 

12.  Given      =h,     to  find  x. 

a-\-x 

±«(lq=  V2b-b'') 

Ans.     x= y    — \ 

V25-^'2 

13.  Given  ^       ^  to  find  a;  and  y 

i  xy=z  6  >  ^ 

(  x=3     or     2     or     — 3±  VY 

Ans.       {  , — 

{  y=2     or     3     or     -3ip  VT. 

14.  Given  the  sum  of  two  numbers  equal  to  a,  and  the  sum  of 

their  cubes  equal  to  c,  to  find  the  numbers 

„      ,  ,.  .  (  X  +7/  =a 

By  the  conditions  {    „ 

(  ar  +  2/-'=c. 

Putting     x=s-\-z,     and    y=s—z,     we  have     a=25, 

C  r'=«='+3^«+352='+r» 


Hence,  by  addition,     x^-]-y^=2s^  -\-6sz^=c 


184 


Whence  2^= — r and    z=±\/  — , 

^ .   /  c-2s'  ,  .    /T^T" 

or  a;=:5±V  — g^— 5  »nd  ?/=5;pV  — g^^ — , 

Or  by  putting  for  s  its  value, 
c 


-|-v/(-3r^)=|-N/- 


4c  — G^ 


a  //  4  \      «  /4c— «^ 

and  ^=_:p^(____j=._:p^___. 

QUESTIONS. 

1.  There  are  two  numbers  whose  difference  is  15,  and  half  their 
product  is  equal  to  the  cube  of  the  lesser  number.  What  are  those 
numbers?  Ans.     3  and  18. 

2.  What  two  numbers  are  those  whose  sum,  multiplied  by  the 
greater,  is  equal  to  77 ;  and  whose  difference,  multiplied  by  the 
lesser,  is  equal  to  12  ? 

Ans.     4  and  7,  or  |  s/2  and  y  ^/2. 

3.  To  divide  100  mto  two  such  parts,  that  the  sum  of  their  square 
roots  may  be  14.  A7is.     64  and  36. 

4.  It  is  required  to  divide  the  number  24  into  two  such  parts,  that 
their  product  may  be  equal  to  35  times  their  difference. 

A71S.     10  and  14. 

5.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their  cubes  is 
152.     What  are  the  numbers  ?  Ans.     3  and  5. 

6.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their  4th  powers 
is  641.     What  are  the  numbers?  A71S.     2  and  5. 

7.  The  sum  of  two  numbers  is  6,  and  the  sum  of  their  5th  pow- 
ers  is  1056.     What  are  the  numbers?  A71S.     2  and  4. 

8.  Two  merchants  each  sold  the  same  kind  of  stuff;  the  second 
sold  3  yards  more  of  it  than  the  first,  and  together,  they  receive  35 


FORMATION  OF  FOWERsO-  •••••«-  185 

crowns.     The  first  said  to  the  second,  I  would  have  received  24 
crowns  for  your  stuff;  the  other  replied,  and  I  would  have  received 
121  crowns  for  yours.     How  many  yards  did  each  of  them  sell  ? 
(  1st  merchant  a;=15  x=5  } 

^"^-  i  2d  .  .  .  y=18  °^-  y=s\' 
9.  A  widow  possessed  13,000  dollars,  which  she  divided  into  two 
parts,  and  placed  them  at  interest,  in  such  a  manner,  that  the  incomes 
from  them  were  equal.  If  she  had  put  out  the  first  portion  at  the 
same  rate  as  the  second,  she  would  have  drawn  for  this  part  360 
dollars  interest,  and  if  she  had  placed  the  second  out  at  the  same 
rate  as  the  first,  she  would  have  drawn  for  it  490  dollars  interest. 
What  were  the  two  rates  of  interest  ? 

Ans.     7  and  6  per  cent. 


CHAPTER  IV. 


Formation  of  Powers,  and  Extraction  of  Roots  of 
any  degree  whatever. 

159.  The  resolution  of  equations  of  the  second  degree  supposes 
the  process  for  extracting  the  square  root  to  be  known  ;  in  like  man- 
ner the  resolution  of  equations  of  the  third,  fourth,  &c.  degree,  re- 
quires that  we  should  know  how  to  extract  the  third,  fourth,  d;c. 
root  of  any  numerical  or  algebraic  quantity. 

It  will  be  the  principal  object  of  this  chapter  to  explain  the  rais- 
ing of  powers,  the  extraction  of  roots,  and  the  calculus  of  radicals. 

Although  any  power  of  a  number  can  be  obtained  from  the  rules 
of  multiplication,  yet  this  power  is  subjected  to  a  certain  law  of  com- 
position which  it  is  absolutely  necessary  to  know,  in  order  to  dedtice 
the  root  from  the  poioer.  Now,  the  law  of  composition  of  the  square 
of  a  numerical  or  algebraic  quantity,  is  deduced  from  the  expression 
for  the  square  of  a  binomial  (Art.  117)  ;  so  likewise,  the  law 
16* 


♦ 

........  r  -Tt  ^ 

186  ALGEBRA. 

.  •  •  • 

t)f  'a  power  'df  any  degree,  is  deduced  from  the  same  power  of 
a  binomial.  We  will  therefore  determine  the  development  of  any 
power  of  a  binomial. 

160.  By  multiplying  the  binomial  x-\-a  into  itself  several  times 
the  following  results  are  obtauied  ; 

(x+a)=x+a, 

{x+ay=x'^+2ax+a^, 

{x+af=x^+3a3r'+Sa''x+a^ 

(a; + a)* = a* + 4aa;' + 6a  V + 4a='a; + «*, 

(a; +a)5 =0,-5 + 5ax'  + 1  OaV  +  lOaV + 5a*a; + a' 

By  inspecting  these  developments  it  is  easy  to  discover  a  law  ac- 
cording to  which  the  exponents  of  x  and  a  decrease  and  increase  in 
the  successive  terms;  it  is  not,  however,  so  easy  to  discover 
a  law  for  the  co-efficients.  Newton  discovered  one,  by  means  of 
which,  any  power  of  a  binomial  can  be  formed,  without  first  obtain- 
ing all  of  the  inferior  powers.  He  did  not  however  explain  the 
course  of  reasoning  which  led  him  to  the  discovery  of  it ;  but  the 
existence  of  this  law  has  since  been  demonstrated  in  a  rigorous 
manner.  Of  all  the  known  demonstrations  of  it,  the  most  elemen- 
tary is  that  which  is  founded  upon  the  theory  of  comhinations.  How- 
ever, as  it  is  rather  complicated,  we  will,  in  order  to  simplify  the  ex- 
position  of  it,  begin  by  resolving  some  problems  relative  to  combi- 
nations, from  which  it  will  be  easy  to  deduce  the  formula  for  the  hi- 
nomial,  or  the  development  of  any  power  of  a  binomial. 

Theory  of  Permutations  and  Comhinations. 

161.  Let  It  be  proposed  to  determine  the  whole  number  of  ways 
in  which  several  letters,  a,  b,  c,  d,  &c.  can  be  written  one  after  the 
other.  The  results  corresponding  to  each  change  in  the  position  of 
any  one  of  these  letters,  are  called  permutations. 

Thus,  the  two  letters  a  and  b  furnish  the  two  permutations  ab 
and  ba. 


PERMUTATIONS  AND  COMBINATIONS.  187 


In  like  manner,  the  three  letters  a,  b,  c,  furnish 
six  permutations. 


'  abc 
acb 
cab 
bac 
bca 
^  cba 

Permutations,  are  the  results  obtained  by  writing  a  certain  number 
of  letters  one  after  the  other,  in  every  possible  order,  in  such  a  man. 
ner  that  all  the  letters  shall  enter  into  each  result,  and  each  letter 
enter  but  once. 

Problem  1.  To  determine  the  number  of  -permutations  of  which 
n  letters  are  susceptible. 

In  the  first  place,  two  letters  a  and  b  evidently  (  ab 

give  two  permutations.  (  ba 

Therefore,  the  number  of  permutations  of  two  letters  is  1 X  2« 

Take   the  three  letters  a,  h,  and  c.     Reserve  r  c 

either  of  the  letters,  as  c,  and  permute  the  other  two,  <    ab 

giving         o ,  \    ba 

Now,  the  third  letter  c  may  be  placed  before  ab, 
between  a  and  b,  and  at  the  right  of  ab  ;  and  the 
same  for  ba  :  that  is,  in  one  of  the  first  permutations 
the  reserved  letter  c  may  have  three  different  places, 
giving  three  'permutations.  Now,  as  the  same  may 
be  shown  for  each  of  the  first  permutations,  it  fol- 
lows  that  the  whole  number  of  permutations  of  three 
letters  will  be  expressed  by  1x2x3. 

If  now,  a  fourth  letter  d  be  introduced,  it  can  have  four  places  in 
each  of  the  six  permutations  of  three  letters  :  hence  all  the  per- 
mutations of  four  letters  will  be  expressed  by  1x2x3x4. 

In  general,  let  there  be  n  letters  a,  b,  c,  &c.  and  suppose  the  total 
number  of  permutations  of  n— 1  letters  to  be  known;  and  let  Q 
denote  that  number.  Now,  in  each  of  the  n— 1  permutations  the 
reserved  letter  may  have  n  places,  giving  n  permutations  :  hence, 


cab 
acb 
abc 
cba 
bca 
bac 


xOO  ALGEBRA. 

when  it  is  so  placed  in  all  of  them,  the  number  of  permutations  will 
be  expressed  by  Qxn. 

Let  n=2.  Q  will  then  denote  the  number  of  permutations  that 
can  be  made  with  a  single  letter;  hence  Q=l,  and  in  this  particu- 
lar case  we  have  Q  X  n~  1x2. 

Let  7i='6.  Q  will  then  express  the  number  of  permutations  of 
3  —  1  or  2  letters,  and  is  equal  to  1x2.  Therefore  Qx^  is  equal 
to  1x2x3. 

Let  71=4:.  Q  hi  this  case  denotes  the  number  of  permutations 
of  3  letters,  and  is  equal  to  1x2x3.  Hence,  Qxn  becomes 
1X2x3x4,  and  similarly  when  there  are  more  letters. 

162.  Suppose  we  have  a  number  m,  of  letters  a,  I,  c,  d,  &c.,  if 
they  are  written  one  after  the  other,  2  and  2,  3  and  3,  4  and  4  .  .  . 
in  every  possible  order,  in  such  a  manner,  however,  that  the  num. 
bar  of  letters  in  each  result  may  be  less  than  the  number  of  given 
letters,  we  may  demand  the  whole  number  of  results  thus  obtamed. 
These  results  are  called  arrange7nenls. 

Thus  ah,  ac,  ad,  .  .  .  ba,  he,  hd,  .  .  .  ca,  ch,  cd,  .  .  .  are  arrange- 
t)ie7its  of  7)1  letters  taken  2  and  2,  or  in  sets  6f  2  each. 

In  like  manner,  ahc,  ahd,  .  ,  .  hac,  bad,  .  .  .  ach,  acd,  .  .  .  are  ar- 
rangemeTits  taken  in  sets  of  3. 

Arrangements,  are  the  results  obtained  by  writing  a  number  m  of 
letters  one  after  the  other  in  every  possible  order,  in  sets  of  2  and 
2,  3  and  3,  4  and  4  .  .  .  n  and  n  ;  m  being  >n :  that  is,  the  num- 
ber of  letters  in  each  set  being  less  than  the  whole  number  of  letters 
considered.  However,  if  we  suppose  n=?n,  the  arrangeme7its  taken 
n  and  n,  will  become  simple  per7nutations. 

Problem  2.  Having  give 71  a  number  m  of  letters  a,  b,'C,  d  .  .  ., 
to  determine  the  total  7iumber  of  arra7igements  that  may  be  for7ned  of 
them  by  taki7ig  them  n  at  a  time ;  m  being  supposed  greater  than  n. 

Let  it  be  proposed,  in  the  first  place,  to  arrange  the  three  letters 
a,  h,  and  c  in  sets  of  two  each. 


PERMUTATIONS  AND  COMBINATIONS.  189 

First,  arrange  the  letters  in  sets  of  one  each,  in  which  r  a 

case  we  say  there  are  two  letters  reserved  :  the  reserved  }    b 

letters  for  either  arrangement,  being  those  which  do  not  (   c 

enter  it. 

ah 
Now,  to  any  one  of  the  letters,  as  a,  annex,  in  suc- 
cession, the  reserved  letters  b  and  c :  to  the  second  ar- 
rangement  b,  annex  the  reserved  letters  a  and  c ;  and 
to  the  third  arrangement  c,  annex  the  reserved  letters  a 
and  b  :  this  gives 

Hence,  we  see,  that  the  arrangements  of  three  letters  taken  two  in 
a  set,  loill  be  equal  to  the  arrangements  of  the  same  number  of  letters 
taken  one  in  a  set,  multiplied  by  the  number  of  reserved  letters. 

Let  it  be  required  to  form  the  arrangement  of  four  letters, 
a,  b,  c,  and  d,  taken  3  in  a  set. 

First,  arrange  the  four  letters  two  in  a  set :  there  will  r  ab 

then  be  two  reserved  letters.  Take  one  of  the  sets  and 
write  after  it,  in  succession,  each  of  the  reserved  letters  : 
we  shall  thus  form  as  many  sets  of  three  letters  each  as 
there  are  reserved  letters  ;  these  sets  differing  from  each 
other  by  at  least  the  last  letter.  Take  another  of  the 
first  arrangements,  and  annex  in  succession  the  reserved 
letters  ;  we  shall  again  form  as  many  different  arrange- 
ments, as  there  are  reserved  letters.  Do  the  same  for 
all  of  the  first  arrangements,  and  it  is  plain,  that  the  whole 
number  of  arrangements  which  will  be  formed,  of  four 
letters,  taken  3  and  3,  will  be  equal  to  the  arrangements  of 
the  same  letters,  taken  two  in  a  set,  multiplied  by  the  num- 
ber of  reserved  letters. 

In  order  to  resolve  this  question  in  a  general  manner,  suppose  the 
total  number  of  arrangements  of  the  m  letters  taken  n— 1  in  a  set 
to  be  known,  and  denote  this  number  by  P. 

Take  any  one  of  these  arrangements,  and  annex  to  it  each  of 
the   reserved   letters,   of  which   the   rumber   is    m—{n—\),    or 


ba 
ac 
ca 
ad 
da 
be 
cb 
hd 
dh 
cd 
L  dc 


190  ALGEBRA. 

?M  — n  +  l  ;  it  is  evident,  that  we  shall  thus  form  a  number  m— n+1 
of  arrangements  of  n  letters,  differing  from  each  other  by  the  last  let- 
ter. Now  take  another  of  the  arrangements  of  w—  1  letters,  and  an- 
nex to  it  each  of  the  m—n+l  letters  which  do  not  make  a  part  of 
it;  we  again  obtain  a  number  m—n-\-\  of  arrangements  of  n  let- 
ters, differing  from  each  other,  and  from  those  obtained  as  above,  at 
least  in  the  disposition  of  one  of  the  n— 1  first  letters.  Now,  as 
we  naay  in  the  same  manner  take  all  the  P  arrangements  of 
the  m  letters,  taken  n—\  in  a  set,  and  annex  to  them  successively 
the  m— n  +  1  other  letters,  it  follows  that  the  total  number  of  ar- 
rangements of  m  letters  taken  n  in  a  set,  is  expressed  by 
P{in-n-\-\). 

To  apply  this  to  the  particular  cases  of  the  number  of  arrange- 
ments  of  m  letters  taken  2  and  2,  3  and  3,  4  and  4,  make  n=2, 
whence  w— n+l=wi— 1;  P  will  in  this  case  express  the  total  num- 
ber of  arrangements,  taken  2—1  and  2—1,  or  1  and  1,  and  is  con- 
sequently equal  to  m  ;  therefore  the  formula  becomes  m{m—\). 

Let  n= 3,  whence  m— n  +  l=m— 2;  P  will  then  express  the 
number  of  arrangements  taken  2  and  2,  and  is  equal  to  m{m—\) ; 
therefore  the  formula  becomes  m{m—\)  (m— 2). 

Again,  take  7i=4,  whence  m— n+l=m— 3  ;  P  will  express  the 
number  of  arrangements  taken,  3  and  3,  or  is  equal  to 

7«(?H— 1)  (m  — 2) : 
therefore  the  formula  becomes 

m{jn  —  \)  (/n  — 2)  (?h  — 3). 

Remark.  From  the  manner  in  which  the  particular  cases  have 
been  djeduced  from  the  general  formula,  we  may  conclude  that  it 
reduces  to 

m{ni—\)  (ot— 2)  (wi— 3)  ....  (m  — «  +  l)  ; 

that  is,  it  is  composed  of  the  product  of  the  n  consecutive  numbers 
comprised  between  m  and  rn— n  +  1,  inclusively. 


PERMUTATIONS  AND  COMBINATIONS.  191 

From  this  formula,  that  of  the  preceding  Art.  can  easily  be  de- 
duced, viz.  the  development  of  the  value  of  Qx^- 

For,  we  see  that  the  arrangements  become  permutations  when  the 
number  of  letters  composing  each  arrangement  is  supposed  equal 
to  the  total  number  of  letters  considered. 

Therefore,  to  pass  from  the  total  number  of  arrangements  of  m 

letters,  taken  n  and  n,  to  the  number  of  permutations  of  n  letters, 

it  is  only  necessary  to  make  m=?i  in  the  above  development,  which 

gives 

n(n-l)  (n-2)  («-3) 1. 

By  reversing  the  order  of  the  factors,  observing  that  the  last  is 
1,  the  next  to  the  last  2,  which  is  preceded  by  3  .  .  .,  it  becomes 

1,  2,  3,  4 (^-2)  {n-l)n, 

for  the  development  of  Q  X  ^^' 

This  is  nothing  more  than  the  series  of  natural  numbers  compris- 
ed  between  1  and  n,  inclusively. 

163.  When  the  letters  are  disposed,  as  in  the  arrangements,  2 
and  2,  3  and  3,  4  and  4,  &c.,  it  may  be  required  that  no  two  of  the 
results,  thus  formed,  shall  be  composed  of  the  same  letters,  in  which 
case  the  products  of  the  letters  will  be  different ;  and  we  may  then 
demand  the  whole  number  of  results  thus  obtained.  In  this  case, 
the  results  are  called  combinations. 

Thus,  ai,  ac,  be,  .  .  .  ad,  bd,  .  .  .  are  combinations  of  the  letters 
taken  2  and  2. 

In  like  manner,  abc,  abd,  .  .  .  acd,  bed  .  .  .  are  combinations  of 
the  letters  taken  3  and  3. 

Combinations,  are  arrangements  in  which  any  two  will  differ  from 
each  other  by  at  least  one  of  the  letters  lohich  enter  them. 

Hence,  there  is  an  essential  difference  in  the  signification  of  the 
words,  permutations,  arrangements,  and  combinations. 

Problem  3.  To  determine  the  total  number  of  different  combina- 
tions that  can  be  formed  of  m  letters,  taken  n  in  a  set. 

Let  X  denote  the  total  number  of  arrangements  that  can  bo 
formed  of  m  letters,  taken  n  and  n  :  F  the  number  of  permutations 


192  ALGEBRA. 

of  n  letters  ;  and  Z  the  total  number  of  different  coinUnations  taken 
«  and  11. 

It  is  evident,  that  all  the  possible  arrangements  of  m  letters,  taken 
w  at  a  time,  can  be  obtained,  by  subjecting  the  n  letters  of  each  of 
the  Z  combinations,  to  all  the  permutations  of  which  these  letters 
are  susceptible.  Now  a  single  combination  of  n  letters  gives,  by 
hypothesis  Y  permutations  ;  therefore  Z  combinations  will  give 
Yx  Z  .  .  .  arrangements,  taken  n  and  n ;  and  as  X  denotes  the 
total  number  of  arrangements,  it  follows  that  the  three  quantities 

X 
X,  Y,  and  Z,  give  the  relations  X=  YxZ;  whence     Z=^. 

But  we  have  (Art.  162),         X=P(m-n+l) 

and  (Art.  161),  Y=Qxn. 

P(m—n  +  l)     P      m—n+\ 

Therefore,         Z=  ^—-: ' = — X . 

QXn  Q  n 

Since  P  expresses  the  total  number  of  arrangements,  taken  n— 1 
and^i— 1,  and   Q  the  number  of  permutations  of  w— 1  letters,  it 

P 

follows  that     —     expresses  the  number  of  different  combinations 

of  m  letters  taken  n— 1  and  n— 1. 

To  apply  this  to  the  particular  case  of  combinations  of  m  letters 
taken  2  and  2,  3  and  3,  4  and  4  .  .  . 

P 

Make  n=2,  in  which  case     —     expresses  the  number  of  com- 

binations  of  the  letters  taken  2—1   and  2—1   or  1  and  1,  and  is 
equal  to  m ;  the  above  formula  becomes 

m  —  \  m{m—\) 


mx- 


2  1.2 


P 

Let  n=S,    —     will  express  thcvnumber  of  combinations  taken 

7w(m— 1) 
2  and  2,  and  is  equal  to     — — —  ;     and  the  formula  becomes 

m{m—l)  (m—2) 
1.2.3  • 


BliVOMIAL  THEOREM. 


193 


In  like  manner,  we  would  find  the  number  of  combinations  of 


letters  taken  4  and  4,  to 


??i(m— 1)  (m—2)  (m  — 3) 


1.2.3.4 


;  and  in  ge- 


neral, the  number  of  combinations  of  m  letters  taken  n  and  n,  is  ex- 
pressed  by 

m(m—l)  {m  —  2)  (m—S)  .  .  .  (m— n  +  1) 
1.2.3.4  .  .  .  {n-l)7n  ' 

which  is  the  development  of  the  expression 
P{m—7i+l) 
Qxn~' 


Demonstration  of  the  Binomial  Theorem. 

164.  In  order  to  discover  more  easily  the  law  for  the  develop- 
ment of  the  mth  power  of  the  binomial  x-}-a,  we  will  observe  the 
law  of  the  product  of  several  binomial  factors  x+a,  x-\-bf  x+c, 
x-\-d  .  .  .  oi^  which  the  first  term  is  the  same  in  each,  and  the  se- 
cond terms  different. 

X  +  a 

X  +  b 


1st.  product 


2d. 


3d. 


x^  +  a 

+  i 

X   -\-  c 

x'  +  a 

+  h 

+  c 

X  -{■  d 


X  +  ab 


X?  +  ab 
+  ac 
+   be 


X  +  abc 


a 
+  b 
+  c 
+  d 


x^  +  (lb 

+  ac 

4-  ad 

+  be 

+  bd 

+  cd 
17 


x^  +  abc 
+  ahd 
+  acd 
+   bed 


X  +  c-bcd 


194  ALGEBRA. 

From  these  products,  obtained  by  the  common  rule  for  algebraic 
multiplication,  we  discover  tlie  following  laws  : 

1st.  With  respect  to  the  exponents  ;  the  exponent  of  x,  in  the  first 
term,  is  equal  to  the  number  of  binomial  factors  employed.  In  the 
following  terms,  this  exponent  diminishes  by  unity  to  the  last  term, 
where  it  is  0. 

2d.  With  respect  to  the  co-efRcients  of  the  different  powers  of  a;; 
that  of  the  first  term  is  unity  ;  the  co-efficient  of  the  second  term  is 
equal  to  the  sum  of  the  second  terms  of  the  binomials  ;  the  co-effi- 
cient  of  the  third  term  is  equal  to  the  sum  of  the  products  of  the 
different  second  terms  taken  two  and  two ;  the  co-efficient  of  the 
fourth  term  is  equal  to  the  sum  of  their  different  products  taken 
three  and  three.  Reasoning  from  analogy,  we  may  conclude  that 
the  co-efficient  of  the  term  which  has  n  terms  before  it,  is  equal  to 
the  sum  of  the  different  products  of  the  m  second  terms  of  the  bi- 
nomials  taken  71  and  n.  The  last  term  is  equal  to  the  continued  pro- 
duct of  the  second  terms  of  the  binomials. 

In  order  to  be  certain  that  this  law  of  composition  is  general,  sup- 
pose that  it  has  been  proved  to  be  true  for  a  number  m  of  binomials  ; 
let  us  see  if  it  be  true  when  a  new  factor  is  introduced  into  the  pro- 
duct. 

For  this  purpose,  suppose 
a?--hAa;-"-*+Bx'"-2+ar"'-^  .  .  .  +Ma;— "+i+N-r"'-'^-f  .  .  .  +U, 
to  be  the  product  of  ?«  binomial  fectors,  Nx"—"  representing  the 
term 'which  has  n  terms  before  it,  and  Ma;"'-"+^  that  which  immedi- 
diately  precedes. 

Let  x+K  be  the  new  factor,  the  product  when  arranged  according 
to  the  powers  of  x,  will  be 

+kI    +ak1      +bkI  +mkI        +UK. 

From  which  we  perceive  that  the  law  of  the  exponents  is  evident- 
ly  the  same. 

With  respect  to  the  co-efficients,  1st.  That  of  the  first  term  is 


BINOMIAL  THEOREM.  195 

ttnity.  2d.  A+K,  or  the  co-efficient  of  x'",  is  also  the  sum  of  the 
second  terms  of 'the  m+1  binomials. 

3d.  B  is  by  hypothesis  the  sum  of  the  difTerent  products  of  the 
second  terms  of  the  m  binomials,  and  A.K  expresses  the  sum  of  the 
products  of  each  of  the  second  terms  of  the'  m  first  binomials,  by 
the  new  second  term  K ;  therefore  B-^-AK  is  tJie  sum  of  the  dif- 
f event  products  of  the  second  terms  of  the  m+1  binomials,  taJcen  two 
and  two. 

In  general,  since  N  expresses  the  sum  of  the  products  of  the  se- 
cond  terms  of  the  m  first  binomials,  taken  n  and  n  ;  and  as  3IK  re- 
presents the  sum  of  the  products  of  these  second  terms,  taken  n— 1 
and  n— 1,  multiplied  by  the  new  second  term  K,  it  follows  that 
N-\-]MK,  or  the  co-efficient  of  the  term  which  has  n  terms  before 
it,  is  equal  to  the  sum  of  the  difierenl  products  of  the  second  terms 
of  the  m+1  binomials,  taken  n  and  n.  The  last  term  is  equal  to 
the  contmued  product  of  the  m+1  second  terms. 

Therefore,  the  law  of  composition,  supposed  true  for  a  number 
m  of  binomial  factors,  is  also  true  for  a  number  denoted  by  m  +  1. 
It  is  therefore  general. 

Let  us  suppose,  that  in  the  product  resulting  from  the  multiplica- 
tion  of  the  m  binomial  factors,  x-\-a,  x-\-b,  x-\-c,  x-\-d  ...  we  make 
a=b=c=d  .  .  .,  the  indicated  expression  of  this  product,  {x-\-a) 
(.T+&)  (x+c),  will  be  changed  into  (x+a)"'.  With  respect  to  its  de- 
velopment, the  co-efficients  being  a+5+c  +  <Z. . .,  fl5+ac+acJ+.  .  ., 
abc-\-abd-\-acd  .  .  .,  the  co-efficient  of  x'""',  or  a  +  Z»+c+(Z  .  .  ., 
becomes  a-{-a-\-a-}-a-{-  .  .  .,  that  is,  a  taken  as  many  times  as  there 
are  letters  a,  b,  c  .  .  .,  and  is  therefore  equal  to  7na.  The  co-effi- 
cient of  x'""^^  or  ab-{-ac-{-ad-\-  .  .  .,  reduces  to  a^-{-a"-\-a^  .  .  .,  or 
to  a^  taken  as  many  times  as  we  can  form  different  combinations  with 

m— 1 
m  letters,  taken  two  and  two,  or  to  m  .  — — — a^.     (Art.  163). 

The  co-efHcient  of  x'""^  reduces  to  the  product  of  a^,  multiplied 


1 96  ALGEBRA. 

by  the  number  of  different  combinations  of  m  letters,  taken  3  and 

_  m— 1     7?i  —  2 

3»  or  to  771  .  — — -a",  &c. 

In  general,  if  the  term,  which  has  7i  terms  before  it,  is  denoted  by 
JSx'"-'",  the  co-efficient,  which  in  the  hypothesis  of  the  second  terms 
being  different,  is  equal  to  the  sum  of  their  products,  taken  n  and 
n,  reduces,  when  all  of  the  terms  are  supposed  equal,  to  a"  multi- 
plied by  the  number  of  different  combinations  that  can  be  made 
with  m  letters,  taken  n  and  7U     Therefore 


P(m~n+1) 
QXn 
From  which  we  have  the  formula 


N=-^^ --^a".     (Art.  163). 


(x-f  a)'"=a;"'-f  wa.r'^-^+m  .  — —a^^-'^s 
m— 1     m—2  P(m—7i  +  l) 

165.  By  inspecting  the  different  terms  of  this  development,  a 
si77iple  law  will  be  perceived,  by  means  of  which  the  co-efficient  of 
any  term  is  formed  from  the  co-efficient  of  the  preceding  term. 

The  co-efficie7it  of  any  ter7n  is  formed  ly  multiply'mg  the  co-effi. 
dent  of  the  preceding  term  hy  the  exponent  of  x  in  that  term,  a7id  di- 
vidi7ig  the  product  by  the  7iimiber  of  ter77is  which  precede  the  required 
ter7n. 

o       .  1       1  ,  P(m  — ?^  +  l) 

For,  take  the  general  term     -^-— ^aV'-" .     This  is  called 

the  general  term,  because  by  making  n=2,  3,  4  .  .  .,  all  of  the 
others  can  be  deduced  from  it.     The  term  which  immediately  pre- 

P  P 

cedes  it,  is  evidently     — a"-^a;'"-''+^   since    —  expresses  the  num- 
ber of  combinations  of  m  letters  taken  n—1  and  n—1.     Here  we 

see  that  the  co-efficient     J^Zl!! 1     [^  equ^l  to  the  co-efficient 

VXn 


BINOMIAL  THEOREM.  197 

P 

-^     which  precedes  it,  multiplied  by  m—n-\-\,  the  exponent  of  «  in 

that  term,  and  divided  by  n,  the  number  of  terms  preceding  the  re- 
quired  term.  This  law  serves  to  develop  a  particular  power,  v/ith- 
out  our  being  obliged  to  have  recourse  to  the  general  formula. 

For  example,  let  it  be  required  to  develop  {x-\-ay.     From  this 
law  we  have, 

After  having  formed  the  first  two  terms  from  the  terms  of  the 
general  formula  x"^ +maaf''''^  -\-  .  .  .,  multiply  6,  the  co-efficient  of 
the  second  term,  by  5,  the  exponent  of  x  in  this  term,  then  divide 
the  product  by  2,  which  gives  15  for  the  co-efficient  of  the  third 
term-  To  obtain  that  of  the  fourth,  multiply  15  by  4,  the  exponent 
of  X'  in  the  third  term,  and  divide  the  product  by  3,  the  number  of 
terms  which  precede  the  fourth,  this  gives  20  ;  and  the  co- efficients 
of  the  other  terms  are  found  in  the  same  way. 
In  like  manner  we  find 

(a;+ay»=a;^''  +  10aa;''  +  45rtV  +  120a^a;^+210aV, 
+252aV  +  210a«x''+120aV+45aV+10a^a;+f/.". 
166.  It  frequently  occurs  that  the  terms  of  the  binomial  are  af- 
fected with  co-efficients  and  exponents,  as  in  the  following  example. 
Let  it  be  required   to   raise  the  binomial  3a^c  —  '2bd  to  the   4th 
power. 

Placing  Sarc—x  and  —2bd=y,  we  have 

(x  +  j^)* = a;* + 4ar'w + 6x^f  +  ^xy"^ +i/ . 
Substituting  for  x  and  y  their  values,  we  have 
{?>cPc-2hciy={2(v'cy+A{Za^cf{-2bd)-\-Q{3cv'cf{-2hdf  + 
4(.3a2c)  {-2hdf  +  {-2bd)\ 
or,  by  performing  the  operations  indicated 

{3c^c-'2My^Q\a^c*-2lQa<'6'bd  +  '2lQa*c'lrd"-QQarchH\ 
+  lQbhl\ 
The  terms  of  the  development  are  alternately  plus  and  minus,  as 
they  should  be,  since  the  second  term  is  — . 
17* 


198  ALGEBRA. 

167.  The  powers  of  any  polynomial  may  easily  be  found  by  the 
binomial  theorem. 

For  example,  raise  a+Zi+c  to  the  third  power. 
First,  put      ....     Z'  +  c=d. 
Then         (a+b+cf={a+df=a'+da^d+Sa(P+d^. 
Or,  by  substituting  for  the  value  of  d, 

{a+b+cy=a'+Sa^+Zab^+P 

Sa^'c  +  Si'^c  +eahc 
+  2ac^+Sbc'' 
+    c\ 
This  expression  is  composed  of  the  cubes  of  the  three  terms,  plus 
three  times  the  square  of  each  term  by  the  first  powers  of  tlie  two 
others,  plus  six  times  the  product  of  all  three  terms.     It  is  easily 
proved  that  this  law  is  true  for  any  polynomial. 

To  apply  the  preceding  formula  to  the  development  of  the  cube 

of  a  trinomial,  in  which  the  terms  are  affected  with  co-efficients  and 

exponents,  designate  each  term  by  a  single  letter,  then  replace  the  let. 

ters  introduced,  by  their  values,  and  perform  the  operations  indicated. 

From  this  rule,  we  will  find  that 

(2a2-4a5+3Z^f=8a«-48a'J+132a^J^-208a='53 
+ 1  QSa'b"  _  1  OSaJs + 21b\ 
The  fourth,  fifth,  &c.  powers  of  any  polynomial  can  be  develop, 
ed  in  a  similar  manner. 

Consequences  of  the  Binomial  Formula. 

168.  First.  The  expression  (x+a)'"  being  such,  that  x  may 
be  substituted  for  a,  and  a  for  x,  without  altermg  its  value,  it  fol- 
lows that  the  same  thing  can  be  done  in  the  development  of  it ; 
therefore,  if  this  development  contains  a  term  of  the  form  Ka"a;'"^'', 
it  must  have  another  equal  to  Kx^a"^'"  or  Ka"'"""a;''.  These  two 
terms  of  the  development  are  evidently  at  equal  distances  from  the 
two  extremes ;  for  the  number  of  terms  which  precede  any  term, 
being  indicated  by  the  exponent  of  a  in  that  term,  it  follows  that 


BINOMIfVL  THEOREM. 


199 


the  term  K^''^;'"""  has  n  terms  before  it ;  and  that  the  term  Ka"'~"oif' 
has  m—n  terms  before  it,  and  consequently  n  terms  after  it,  since 
the  whole  number  of  terms  is  denoted  by  m  +  1. 

Therefore,  in  the  development  of  any  poiver  of  a  hinomiul,  the  co- 
efficients at  equal  distances  from  the  two  extremes  are  equal  to  each 
other. 

Remark.  In  the  terms  Ka^T""-",  Krt™-"af ,  the  first  co-efficient  ex- 
presses  the  number  of  different  combinations  that  can  be  formed  with 
m  letters  taken  n  and  n  ;  and  the  second,  the  number  which  can  be 
formed  when  taken  m—n  and  ?n—n;  we  may  therefore  conclude 
that,  the  number  of  different  combinations  of  m  letters  taken  n  and  n, 
is  equal  to  the  number  of  combinations  of  m  letters  taken  m—n  and 
m—n. 

For  example,  twelve  letters  combined  5  and  5,  give  the  same 
number  of  combinations  as  these  twelve  letters  taken  12  —  5  and 
12  —  5,  or  7  and  7.  Five  letters  combined  2  and  2,  give  the  same 
number  of  combinations  as  five  letters  combined  5—2  and  5—2,  or 
3  and  3. 

169-  Second.  If  in  the  general  formula, 

m — 1 

{x-\-aY^x'''  -{-max'"~^  -\-m — -— a^a;'"""^+,  &c. 

we  suppose  x=l,  0=1,  it  becomes 

m  —  \  711—1         m  —  2 

(1  +  1)-  or  2'"=l+m+m— ^— +7n-— —  .  ~^— +,  &c. 

That  is,  the  sum  of  the  co-efficients  of  the  different  terms  of  the 
formula  for  the  binomial,  is  equal  to  the  mth  power  of  2. 

Thus,  in  the  particular  case 

(x + a)5 = 0,-5  +  ^ax"  + 1  Oa  V + 1  Oa^'ar + ba'x + a\ 
the  sum  of  the  co-efficients  1+5+10  +  104-5+1  is  equal  to  2^  or 
32.     In  the  10th  power  developed,  the  sum  of  the  co-efficients  is 
equal  to  2"  or  1024. 

170.   Third.  In  a  series  of  numbers  decreasing  by  unity,  of  which 


200  ALGEBRA. 

the  first  term  is  m  and  the  last  m—p,  m  and  p  being  entire  numbers, 
the  continued  product  of  all  these  numbers  is  divisible  by  the  con- 
tinued  product  of  all  the  natural  numbers  from  1  to  ^+1  inclu- 
sively. 

Thnt  i«      »<^^-l)  {m-2)  (m-3)  .  .  .  {m-p)    . 

That  IS,      1.2      ^      3      ^      ^  __  (^^^^)    IS  a  whole  num. 

ber.  For,  from  what  has  been  said  in  (Art.  16-3),  this  expression 
represents  the  number  of  different  combinations  that  can  be  formed 
of  m  letters  taken  p+1  and  p+1.  Now  this  number  of  combina- 
tions is,  from  its  nature,  an  entire  number ;  therefore  the  above  ex- 
pression is  necessarily  a  whole  number. 

Of  the  Extraction  of  the  Roots  of  jJ articular  numbers. 

171.  The  third  power  or  cube  of  a  number,  is  the  product  arising 
from  multiplying  this  number  by  itself  twice  ;  and  the  thirds  or  cube 
root,  is  a  number  which,  being  raised  to  the  third  power,  will  produce 
the  proposed  number. 

The  ten  first  numbers  being 

1,  2,   3,    4,     &,      6,       7,      8,       9,       10. 
their  cubes  are         1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

Reciprocally,  the  numbers  of  the  first  line  are  the  cube  roots  of 
the  numbers  of  the  second. 

By  mspecting  these  lines,  we  perceive  that  there  are  but  nine 
perfect  cubes  among  numbers  expressed  by  one,  two,  or  three  figures ; 
each  of  the  other  numbers  has  for  its  cube  root  a  whole  number,  plus 
a  fraction  which  cannot  be  expressed  exactly  by  means  of  unity,  as 
may  be  shown,  by  a  course  of  reasoning  entirely  similar  to  that 
pursued  in  the  latter  part  of  (Art.  118). 

172.  The  difference  between  the  cubes  of  two  consecutive  num- 
bers increases,  when  the  numbers  are  increased. 

Let  a  and  a+1,  be  two  consecutive  whole  numbers ;  we  have 
(rt  +  lf=a^  +  3a'  +  3a  +  l; 
whence  {a-\-lf-a''^'^a'+^a+\. 


EXTRACTION  OF  ROOTS. 


201 


That  is,  the  difference  between  the  cubes  of  tioo  consecutive  whole 
numbers,  is  equal  to  three  times  the  square  of  the  least  number,  plus 
three  times  this  number,  plus  1. 

Thus,  the  difference  between  the  cube  of  90  and  the  cube  of  89, 
is  equal  to  3(89)2  +  3x89  +  1  =  24031. 

173.  In  order  to  extract  the  cube  root  of  an  entire  number,  we 
will  observe,  that  when  the  figures  expressing  the  number  do  not 
exceed  three,  its  root  is  obtained  by  merely  inspecting  the  cubes  of  the 
first  nine  numbers.  Thus,  the  cube  root  of  125  is  5  ;  the  cube  root 
of  72  is  4  plus  a  fraction,  or  is  within  one  of  4  ;  the  cube  root  of 
841  is  within  one  of  9,  since  841  falls  between  729,  or  the  cube  of 
9,  and  1000,  or  the  cube  of  10. 

When  the  number  is  expressed  by  more  than  three  figures,  the 
process  will  be  as  follows.     Let  the  proposed  number  be  103823. 


103.823 

47 

64 

8 

1  398.23 

.  48 

47 

48 

47 

384 

329 

192 

188 

2304 

2209 

48 

47 

18432 

15463 

9216 

8836 

110592 

103823 

This  number  being  comprised  between  1,000,  which  is  the  cube 
of  10,  and  1,000,000,  which  is  the  cube  of  100,  its  root  will  be  ex- 
pressed  by  two  figures,  or  by  tens  and  units.  Denoting  the  tens  by 
a,  and  the  units  by  b,  we  have  (Art.  160), 

{a-^bf=:a?  +  2,a?b  +  2a¥-\-V\ 

Whence  it  follows,  that  the  cube  of  a  number  composed  of  tens 
and  units,  is  equal  to  the  cube  of  the  tens,  plus  three  times  the  product 


202  ALGEBRA. 

of  the  square  of  the  tens  ly  the  units,  plus  three  times  the  product  of 
the  tens  by  the  square  of  the  units,  plus  the  cube  of  the  units. 

This  being  the  case,  the  cube  of  the  tens,  giving  at  least,  thou- 
sands,  the  last  three  figures  to  the  right  cannot  form  a  part  of  it  :  the 
cube  of  the  tens  must  therefore  be  found  in  the  part  103  which  is 
separated  from  the  last  three  figures  by  a  point.  Now  the  root  of 
the  greatest  cube  contained  in  103  being  4,  this  is  the  number  of 
tens  in  the  required  root ;  for  103823  is  evidently  comprised  be- 
tween  (40)=*  or  64,000,  and  (50)=*  or  125,000 ;  hence  the  required 
root  is  composed  of  4  tens,  plus  a  certain  number  of  units  less  than 
ten. 

Having  found  the  number  of  tens,  subtract  its  cube  64  from  103  ; 
there  remains  39,  and  bringing  down  the  part  823,  we  have  39823, 
which  contains  three  times  the  square  of  the  tens  by  the  units,  plus 
the  two  parts  before  mentioned.  Now,  as  the  square  of  a  number 
of  tens  gives  at  least  hundreds,  it  follows  that  three  times  the  square 
of  the  tens  by  the  units,  must  be  found  in  the  part  398,  to  the  left  of 
23,  which  is  separated  from  it  by  a  point.  Therefore,  dividing  398 
by  three  times  the  square  of  the  tens,  which  is  48,  the  quotient  8 
will  be  the  unit  of  the  root,  or  something  greater,  since  398  hun- 
dreds is  composed  of  three  times  the  square  of  the  tens  by  the  units, 
together  with  the  two  other  parts.  We  may  ascertain  whether  the 
figure  8  is  too  great,  by  forming  the  three  parts  which  enter  into 
39823,  by  means  of  the  figure  8  and  the  number  of  tens  4  ;  but  it 
is  much  easier  to  cube  48,  as  has  been  done  in  the  above  table.  Now 
the  cube  of  48  is  110592,  which  is  greater  than  103823 ;  therefore 
8  is  too  great.  By  cubing  47  we  obtain  103823  ;  hence  the  pro- 
posed number  is  a  perfect  cube,  and  47  is  the  cube  root  of  it. 

Remark.  The  units  figures  could  not  be  first  obtained ;  because 
the  cube  of  the  units  might  give  tens,  and  even  hundreds,  and  the 
tens  and  hundreds  would  be  confounded  with  those  which  arise  from 
other  parts  of  the  cube. 


EXTRACTION  OF  ROOTS. 


>03 


Affain,  extract  the  cube  root  of  47954 


47.954 

36 

27 

36 

3^X3==27 

1  209 

37 

36 

37 

47954 

216 

259 

46656 

108 

111 

1298 

1296 

1369 

36 

37 

7776 

9583 

3888 

4107 

46656 


50653 


The  number  47954  being  below  1,000,000,  its  root  contains  only 
two  figures,  viz.  tens  and  units.  The  cube  of  the  tens  is  found  m 
47  thousands,  and  we  can  prove,  as  in  the  preceding  example,  that 
3,  the  root  of  the  greatest  cube  contained  in  47,  expresses  the  tens. 
Subtracting  the  cube  of  3  or  27,  from  47,  there  remains  20  ;  bring- 
ing down  to  the  right  of  this  remainder  the  figure  9  from  the  part 
954,  the  number  209  hundreds,  is  composed  of  three  times  the 
square  of  the  tens  by  the  units,  plus  the  number  arising  from  the 
other  two  parts.  Therefore,  by  forming  three  times  the  square  of 
the  tens,  3,  which  is  27,  and  dividing  209  by  it,  the  quotient  7  will 
be  the  units  of  the  root,  or  something  greater.  Cubing  37,  we  have 
50653,  which  is  greater  than  47954 ;  then  cubing  36,  we  obtain 
46656,  which  subtracted  from  47954,  gives  1298  for  a  remainder. 
Hence  the  proposed  number  is  not  a  perfect  cube ;  but  36  is  its 
root  to  within  unity.  In  fact,  the  difference  between  the  proposed 
number  and  the  cube  of  36,  is,  as  we  have  just  seen,  1298,  which 
is  less  than  3(36)^+3x36  +  1,  for  in  verifying  the  result  we  have 
obtained  3888  for  three  times  the  square  of  36. 

174.  Again,  take  for  another  example,  the  number,  43725658 
containing  more  than  6  figures. 


204 


3^X3= 


35^X3  =  3675 


Rem. 


ALGEBRA. 

43.725.658 

352 

27 

1  167 

35 

352 

35 

352 

43  725 

175 

704 

42  875 

105 
1225 

1760 

8506 

1056 

35 

123904 

43725658 

6125 

352 

43614208 

3675 

247808 

111450 

42875 

619520 

371712 

43614208 

Now  the  required  root  contains  more  than  one  figure,  and  may 
be  considered  as  composed  of  units  and  tens  only,  the  tens  being 
expressed  by  one  or  more  figures. 

Since  the  cube  of  the  tens  gives  at  least  thousands,  it  must  be 
found  in  the  part  which  is  to  the  lefl;  of  the  last  three  figures  658. 
I  say  now  that  if  we  extract  the  root  of  the  greatest  cube  contain- 
ed  in  the  part  43725,  considered  with  reference  to  its  absolute  value, 
we  shall  obtain  the  whole  number  of  tens  of  the  root ;  for  let  a  be 
the  root  of  43725,  to  within  unity,  that  is,  such  that  43725  shall  be 
comprised  between  a^  and  (a  +  1)^  ;  then  will  43725000  be  compre- 
hended between  a?X  1000  and  (a+l)^x  1000  ;  and  as  these  two  last 
numbers  differ  from  each  other  by  more  than  1000,  it  follows  that 
the  proposed  number  itself,  43725658,  is  comprised  between  a^x  1000 
and  (a  4-1)^X1 000  ;  therefore  the  required  root  is  comprised  be- 
tween that  of  a^x  1000,  and  (a  +  l)^X  1000,  that  is,  between  ax  10 
and  (a+l)xlO-  It  is  therefore  composed  of  a  tens,  plus  a  certain 
number  of  units  less  than  ten. 

The  question  is  then  reduced  to  extracting  the  cube  root  of  43725 ; 
but  this  number  having  more  than  three  figures,  its  root  will  con- 


EXTRACTION  OF  ROOTS. 


205 


tain  more  than  one,  that  is,  it  will  contain  tens  and  units.  To  ob- 
tain the  tens,  point  off  the  last  three  figures,  725,  and  extract  the 
root  of  the  greatest  cube  contained  in  43. 

The  greatest  cube  contained  in  43  is  27,  the  root  of  which  is  3  ; 
this  figure  will  then  express  the  tens  of  the  root  of  43725,  or  the 
figure  in  the  place  of  hundreds  in  the  total  root.  Subtracting  the 
cube  of  3,  or  27,  from  43,  we  obtain  16  for  a  remainder,  to  the  right 
of  which  bring  down  the  first  figure  7,  of  the  second  period  725, 
which  gives  167. 

Taking  three  times  the  square  of  the  tens,  3,  which  is  27,  and 
dividing  167  by  it,  the  quotient  6  is  the  unit  figure  of  the  root  of 
43725,  or  something  greater.  It  is  easily  seen  that  this  number  is 
in  fact  too  great ;  we  must  therefore  try  5.  The  cube  of  35  is 
42875,  which,  subtracted  from  43725,  gives  850  for  a  remainder, 
which  IS  evidently  less  than  3  x  (35)^+3x35  +  1.  Therefore,  35 
is  the  root  of  the  greatest  cube  contained  in  43725 ;  hence  it  is  the 
number  of  tens  in  the  required  root. 

To  obtain  the  units,  bring  down  to  the  right  of  the  remainder  850, 
the  first  figure,  6,  of  the  last  period,  658,  which  gives  8506  ;  then 
take  3  times  the  square  of  the  tens,  35,  which  is  3675,  and  divide 
8506  by  it ;  the  quotient  is  2,  which  we  try  by  cubing  352  :  tiiis 
gives  43614208,  which  is  less  than  the  proposed  number,  and  sub- 
tracting it  from  this  number,  we  obtain  111450  for  a  remainder. 
Therefore  352  is  the  cube  root  of  43725658,  to  within  unity. 
Hence,  for  the  extraction  of  the  cube  root  we  have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  three  figures  each,  he. 
ginning  at  the  right  hand  :  the  left  hand  period  will  often  contain  less 
than  three  places  of  figures. 

II.  Seek  the  greatest  cube  in  the  first  period,  at  the  left,  and  set  its 
root  on  the  right,  after  the  manner  of  a  quotient  in  division.  Subtract 
the  cube  of  this  figure  of  the  root  from  the  first  period,  and  to  the  re- 

18 


206 


ALGEBRA. 


mainder  bring  down  the  first  figure  of  the  next  period,  and  call  this 
number  the  dividend. 

III.  Take  three  times  the  square  of  the  root  just  found  for  a  divi. 
sor,  and  see  haw  often  it  is  contained  in  the  dividend,  and  place  the 
quotient  for  a  second  figure  of  the  root.  Then  cube  the  figures  of  the 
root  thus  found,  and  if  their  cube  be  greater  thajUhe  first  two  periods 
of  the  given  number,  diminish  the  last  figure  ;  bulif  it  be  less,  subtract 
it  from  the  first  tioo  periods,  and  to  the  reinainder  bring  down  the  first 
figure  of  the  next  period,  for  a  new  dividend.   ■ 

IV.  Take  three  times  the  square  of  the  whole  root  for  a  new  divi- 
sor,  and  seek  how  often  it  is  contained  in  the  new  dividend :  the  quo- 
tient will  be  the  third  figure  of  the  root.  Cube  the  whole  root  and 
subtract  the  result  from  the  three  first  periods  of  the  given  number, 
and  proceed  in  a  similar  way  for  all  the  periods. 

Remark.  If  any  of  the  remainders  are  equal  to,  or  exceed, 
three  times  the  square  of  the  root  obtained  plus  three  times  this  root, 
plus  one,  the  last  figure  of  the  root  is  too  small  and  must  be  aug- 
mented by  at  least  unity  (Art.  172). 

EXAMPLES, 


1.  V48228544=364. 

2.  V27054036008  =  3002. 

3.  V483249=78,  with  a  remainder  8697; 


4.  '/91632508641  =  4508,  with  a  remainder  20644129  • 

5.  V32977340218432=::  32068. 

To  extract  the  n""  root  of  a  ivhole  number. 
175.  In  order  to  generalize  the  process  for  the  extraction  of  roots, 
we  will  denote  the  proposed  number  by  N,  and  the  degree  of  the 
root  to  be  extracted  by  n.  If  the  number  of  figures  in  iV,  does  not 
exceed  n,  the  root  will  be  expressed  by  a  single  figure,  and  is  ob- 
tained immediately  by  forming  the  n"'  power  of  each  of  the  whole 


EXTRACTION  OF  ROOTS. 


207 


numbers  comprised  between  1  and  10  ;  for  the  n"'  power  of  9  is  the 
largest  perfect  power  which  can  be  expressed  by  n  figures. 

When  N  contains  more  than  n  figures,  there  will  be  more  than 
one  figure  in  the  root,  which  may  then  be  considered  as  composed 
of  tens  and  units.  Designating  the  tens  by  a,  and  the  units  by  b, 
we  have  (Art.  166), 

n—\ 
iV=(a  +  Z»)''=a''  +  na''-V>'+n— ~a"-2^2+,  &,c.  ; 

that  is,  the  proposed  number  contains  the  n*''  power  of  the  tens,  plus 
n  times  the  product  of  the  n  — 1  power  of  the  tens  by  the  units,  plus  a 
series  of  other  parts  which  it  is  not  necessary  to  consider. 

Now,  as  the  »"'  power  of  the  tens  cannot  give  units  of  an  order 
inferior  to  unity  followed  by  n  ciphers,  the  last  n  figures  on  the  right, 
cannot  make  a  part  of  it.  They  must  then  be  pointed  off,  and  the 
root  of  the  greatest  n""  power  contained  in  the  figures  on  the  left 
should  be  extracted ;  this  root  will  be  the  tens  of  the  required  root. 

If  this  part  on  the  left  should  contain  more  than  n  figures,  the  n 
figures  on  the  right  of  it,  must  be  separated  from  the  rest,  and  the 
root  of  the  greatest  n""  power  contained  in  the  part  on  the  left  ex- 
tracted, and  so  on.     Hence  the  following 
RULE. 

I.  Divide  the  number  N  into  periods  of  x\  figures  each,  beginning 
at  the  right  hand  ;  extract  the  root  of  the  greatest  n"'  power  contained 
ill  the  left  hand  period,  and  subtract  tJie  n""  poioer  of  this  figure  from 
the  left  hand  period. 

[I.  Bring  down  to  the  right  of  the  remainder  corresponding  to  the 
first  period,  the  first  figure  of  the  second  period,  and  call  this  number 
the  dividend. 

III.  For?n  the  n— 1  power  of  the  first  figure  of  the  root,  multiply  it 
by  n,  and  see  how  often  the  product  is  contained  in  the  dividend :  the 
quotient  will  be  the  second  figure  of  the  root,  or  something  greater. 

IV.  liaise  the  number  thus  formed  to  the  n'**  power,  then  subtract 
this  result  from  the  two  first  periods,  and  to  the  new  remainder  bring 
down  the  first  figwe  of  the  third  period  :  then  divide  the  number  thus 


20S  ALGEBRA 

formed  by  n  times  the  n— 1  power  of  the  two  figures  of  the  root  al- 
ready found,  and  continue  this  operation  until  all  the  periods  are 
brought  down. 

EXAMPLES. 

Extract  the  4th  root  of  531441. 

53.1441    I  27 
2^=  16 

4X2^=32  I  371 
(27)"=  531441. 

We  first  divide  off,  from  the  right  hand,  the  period  of  four  figures, 
and  then  find  the  greatest  fourth  root  contained  in  53,  the  first 
period  to  the  left,  which  is  2.  We  next  subtract  the  4th  power  of 
2,  which  is  16,  from  53,  and  to  the  remainder  37  we  bring  down  the 
first  figure  of  the  next  period.  We  then  divide  371  by  4  times  the 
cube  of  2,  which  would  give  8  for  a  quotient ;  but  by  raising  28  to 
the  4th  power,  we  discover  that  8  is  too  large,  then  trying  7  we  find 
the  exact  root  to  be  27. 

176.  Remark.  When  the  degree  of  the  root  to  be  extracted  is  a 
multiple  of  two  or  more  numbers,  as  4,  6,  .  .  .  .,  the  root  can  he  oh- 
tuined  by  extracting  the  roots  of  more  simple  degrees,  successively. 
To  explain  this,  we  will  remark  that, 

and  that  in  general  (a'")''=a'"xa'"Xa'"Xa"'  •  •  •  =«"""■  (Art.  13). 
Hence,  the  n'*"  power  of  the  m""  power  of  a  number,  is  equal  to  the 
mn""  power  of  this  number. 

Reciprocally,  the  mn"'  root  of  a  number  is  equal  to  the  n""  root  of 
the  m"'  root  of  this  number,  or  algebraically 

"'V~a=  V    V  a  =  V    Va^ 
For,  let  .  .  .  Vv«=«'>  raising  both  membei-s  to  the  n'*  power 
there  will  result  .  .  .  V^=«'"  ;  for  from  the  definition  of  a  root,  we 
have  (  Vir)''=K. 


EXTRACTION  OF  ROOTS.  209 

Again,  by  raising  both  members  to  the  m"'  power,  we  obtain 
a= (a'")'"  =:«""".  Extracting  the  /»«'"  root  of  both  members,  "'!ya=a' ; 
but  we  already  have  V'^a=a' ;  hence  "■;/«=  V'^a. 

In  a  similar  manner  we  might  find   v  a=  \/    Va. 
By  this  method  we  find  that 

V256  =\/  V  256  =r  'V/16  =  4  ; 


V  2985984  =  \/   ^  2985984  =  Vl728= 

V  1771561  =V     V  1771561  =11  ; 


V  1679616  =  Vl296  =  ' 

Remark.  Although  the  successive  roots  may  be  extracted  in 
any  order  whatever,  it  is  better  to  extract  the  roots  of  the  lowest 
degree  first,  for  then  the  extraction  of  the  roots  of  the  higher  de- 
grees,  which  is  a  more  complicated  operation,  is  effected  upon  num. 
bers  containing  fewer  figures  than  the  proposed  number. 

Extraction  of  Roots  by  approximation. 

177.  When  it  is  required  to  -extract  the  n""  root  of  a  number 
which  is  not  a  perfect  power,  the  method  of  (Art.  175),  will  give 
only  the  entire  part  of  the  root,  or  the  root  to  within  unity.  As  to 
the  fraction  which  is  to  be  added,  in  order  to  complete  the  root,  it 
cannot  be  obtained  exactly,  but  we  can  approximate  as  near  as  we 
please  to  the  required  root. 

Let  it  be  required  to  extract  the  n"'  root  of  the  whole  number  a, 

to  within  a  fraction     — ;    that  is,  so  near  it,  that  the  error  shall  be 
P 

1 
less  than     — . 
P 

(ly^pT 

We  M'ill  observe  that  a  can  be  put  under  the  form .     If 

p" 

18* 


210  ALGEBRA. 

axp" 


we  denote  the  root  of  ap"  to  within  unity,  by  r,  the  number 


P" 


or  a,  will  be  comprehended  between     —     and     —  ;     there- 

fore    the      Va    will    be    comprised    between    the    two   numbers, 

r  r+1  r 

—     and     .     Hence     —     is  the  required  root,  to  within  the 

p  p  P 

fraction      — . 
P 
Hence,  to  extract  the  root  of  a  whole  number  to  within  a  fraction 

— ,  multiply  the  number  by  p" ;  extract  the    n""  root  of  the  product  to 

within  unity,  and  divide  the  result  by  p. 

178.  Again,  suppose  it  is  required  to  extract  the  n""  root  of  the 

fraction  -r-. 

0 

Multiply  each  term  of  the  fraction  by 

a      ab"-^ 
b"-^ ;  It  becomes    -7-=    ,     . 
b        ft" 

Let  r  denote  the  n""  root  of  aft"-',  to  within  unity; 

or  — ,  will  be  comprised  between  -j-    and    — — — 

Therefore,  after  having  made  the  denominator  of  the  fraction  a  per- 
fect power  of  the  n'*"  degree,  extract  the  n^^  root  of  the  numerator,  to 
within  unity,  and  divide  ilie  result  by  the  root  of  the  new  denominator. 
When  a  greater  degree  of  exactness  is  required  than  that  indi- 
cated by  -^,  extract  the  root  of  aS"-'  to  within  any  fraction  —  ; 
•'     b  p 

and  designate  this  root  by    — .     Now,  since    —  is  the  root  of  the 
&  P  P 

1  / 

numerator  to  within  — ,  it  follows,  that     -7-     is  the   true   root  of 

p'  bp 

1 

the  fraction  to  within    -r-. 
bp 


EXTRACTION  OF  ROOTS.  211 

179.  Suppose  it  is  required  to  extract  the  cube  root  of  15,  to 

within  — .      We  have    15xl2'=15x  1728=25920.      Now   the 

cube  root  of  25920,  to  within  unity,  is  29  ;  hence  the  required  root 

29  5 

'^     1^     ''    'l2- 

I  • 

Again,  extract  the  cube  root  of  47,  to  within  — . 

We  have    47x20^=47x8000=376000.     Now  the  cube  root 

, —    72       12 
of  376000,   to  within  unity,  is  72;    hence     3/47=— =3—,     to 

1 
withm  -. 

Fmd  the  value  of  V25  to  within  0,001. 

To  do  this,  multiply  25  by  the  cube  of  1000,  or  by  1000000000, 
which  gives  25000000000.  Now,  the  cube  root  of  this  number,  is 
2920  ;  hence    V25= 2,920  to  within  0,001. 

In  general,  in  order  to  extract  the  cube  root  of  a  whole  number  to 
within  a  given  decimal  fraction,  annex  three  times  as  many  ciphers  to 
the  number,  as  there  are  decimal  places  in  the  required  root ;  extract 
the  cube  root  of  the  number  thus  formed  to  within  unity,  and  point  off 
from  the  right  of  this  root  the  required  number  of  decimals. 

180.  We  will  now  explain  the  method  of  extracting  the  cube  root 
of  a  decimal  fraction.  Suppose  it  is  required  to  extract  the  cube 
root  of  3,1415. 

As  the  denominator  10000,  of  this  fraction,  is  not  a  perfect  cube, 
it  is  necessary  to  make  it  one,  by  multiplying  it  by  100,  which 
amounts  to  annexing  two  ciphers  to  the  proposed  decimal,  and  we  have 
3,141500.  Extract  the  cube  root  of  3141500,  that  is,  of  the  num- 
ber considered  independent  of  the  comma,  to  within  unity ;  this 

gives  146.      Then  divide  by  100,  or    VlOOOOOO,  and  we  find 
V3,1415=l,46  to  within  0,01. 


212  ALGEBRA. 

Hence,  to  extract  the  cube  root  of  a  decimal  number,  we  have 
the  following 

RULE 

Annex  ciphers  to  the  decimal  part,  if  necessary,  until  it  can  be 
divided  into  exact  periods  of  three  figures  each,  ohserving  that  the 
number  of  periods  must  be  made  equal  to  the  number  of  decimal 
places  required  in  the  root.  Then,  extract  the  root  as  in  entire  num. 
bers,  and  pioint  off  as  many  places  for  decimals  as  there  are  periods 
in  the  decimal  part  of  the  number. 

To  extract  the  cube  root  of  a  vulgar  fraction  to  within  a  given 
decimal  fraction,  the  most  simple  method  is  to  reduce  the  proposed 
fraction  to  a  decimal  fraction,  continuing  the  operation  until  the  num. 
ber  of  decimal  places  is  equal  to  three  times  tlie  number  required 
in  the  root.  The  question  is  then  reduced  to  extracting  the  cube 
root  of  a  decimal  fraction. 

181.  Suppose  it  is  required  to  find  the  sixth  root  of  23,  to 
within  0,01. 

Applying  the  rule  of  Art.  177  to  this  example,  we  multiply  23 
by  100%  or  annex  twelve  ciphers  to  23,  extract  the  sixth  root  of  the 
number  thus  formed  to  within  unity,  and  divide  this  root  by  100,  or 
point  off  two  decimals  on  the  right. 

In  this  way  we  will  find  that     V23r=l,68,  to  within  0,01. 

EXAMPLES. 

1.  Find  the  V473    to  within  J^.  Ans.     7^. 

2.  Find  the  V79      to  within  ,0001.  Ans.     4,2908. 

3.  Find  the  Vl3      to  within  ,01.  Ans.     1,53. 

4.  Find  the  V3,00415  to  within  ,0001.  Ans.     1,4429. 


5.  Find  the    VOjOOlOl  to  within  ,01.  Ans.     0,10. 

to  within  ,001.  Ans.     0,824. 


EXTRACTION  OF  ROOTS.  213 

Formation  of  Powers    and  Extraction  of  Roots  of  A/gebraic 
Quantities.      Calculus  of  Radicals. 
We  will  first  consider  monomials. 

182.  Let  it  be  required  to  form  the  fifth  power  of  2a^^.  We 
have 

i^o'l/f  ==  2a?¥  X  'ia?!)"  X  2a^Z^  x  2a^¥  X  2a^^^ 
from  which  it  follows,    1st.   That  the  co-efficient  2  must  be  multi- 
plied by  itself  four  times,  or  raised  to  the  fifth  power.     2d.  That 
each  of  the  exponents  of  the  letters  must  be  added  to  itself  four 
times,  or  multiplied  by  5. 

Hence,  (2a='Zr')5=2^a^  ^  ^P>^^='^2a^^b^''. 

In  like  manner,         {Sa''U'cf=&'.a'>^W>^''c''=bl2a^h^c'. 

Therefore,  in  order  to  raise  a  monomial  to  a  given  power,  raise 
the  co-efficient  to  this  poioer,  and  multiply  the  eoqionent  of  each  of  the 
letters  by  the  exponent  of  the  power. 

Hence,  reciprocally,  to  extract  any  root  of  a  monomial,  1st. 
Extract  the  root  of  the  co-efficient.  2d.  Divide  the  exponent  of  each 
letter  hy  the  index  of  the  root. 

V64a^^=  4a='<5c2  ;  Vl6a«^''V= ^a'^Pc. 
From  this  rule,  we  perceive,  that  in  order  that  a  monomial  may 
be  a  perfect  power  of  the  degree  of  the  root  to  be  extracted,  1st. 
its  co-efficient  must  be  a  perfect  power ;  and  2d.  the  exponent  of 
each  letter  must  be  divisible  by  the  index  of  the  root  to  be  extracted. 
It  will  be  shown  hereafter,  how  the  expression  for  the  root  of  a 
quantity  which  is  not  a  perfect  power  is  reduced  to  its  simplest 
terms. 

183.  Hitherto,  we  have  paid  no  attention  to  the  sign  with 
which  the  monomial  may  be  affected  ;  but  if  we  observe,  that  what- 
ever may  be  the  sign  of  a  monomial,  its  square  is  always  positive, 
and  that  every  power  of  an  even  degree,  2n,  can  be  considered  as 
the  n**  power  of  the  square,  that  is,  a^'=(o^)",  it  will  follow  that, 


214  ALGEBRA. 

every  power  of  a  quantity,  of  an  even  degree,  whether  positive  or 
negative,  is  essentially  positive. 

Thus,  {±2aWcy=  +  lGaWc\ 

Again,  as  a  power  of  an  uneven  degree,  2nH-l,  is  the  product 
of  a  power  of  an  even  degree,  2»,  by  the  first  power,  it  follows 
that,  every  power  of  an  uneven  degree  cf  a  monomial,  is  affected  with 
the  same  sign  as  the  monomial. 

Hence,  (+4a2^)='=+64rtW ;   {^-^a^bf^-QW^P. 

From  this  it  is  evident,  1st.  That  when  the  degree  of  the  root  of 
a  monomial  is  uneven,  the  root  .will  be  affected  with  the  same  sign 
as  the  quantity. 

Therefore, 

V+8a^  =  +  2«  ;      V-Sa^^  - 2a ;      V-32a"6==  -  2a^^. 

2d.  When  the  degree  of  the  root  is  even,  and  the  monomial  a 
positive  quantity,  the  root  is  affected  either  with  +  or  — . 

Thus,  V81a^=±3aZ'=';      V64^^=±2a^ 

3d.  When  the  degree  of  the  root  is  even,  and  the  monomial  nega- 
tive, the  root  is  impossible  ;  for,  there  is  no  quantity  which,  raised  to 
a  power  of  an  isven  degree,  can  give  a  negative  result.  Therefore, 
V— a,  V  —  b,  V  —  c,  are  symbols  of  operations  which  it  is 
impossible  to  execute.  They  are,  like  V —a,  V —h,  imagina- 
ry expressions  (Art.  126). 

184.  In  order  to  develop  {a+y-{-z\^,  we  will  place  y-\-z=u,  and 
we  have 

(a +«)='= a^*  +  Sa^M  +  3a?i^ + M^ 
or  by  replacing  u  by  its  value,  y+z 

{a+y+zf=.a^  +  Za\y+z)  +  2a{y+zf+{yJrzf, 
or  performing  the  operations  indicated 

{a+y+zf=a'+2,a^y-\-Za\  +  ^af+Qayz-\-Zaz'^+f-\-^fz-{- 
Syz'^+z^. 

When  the  polynomial  is  composed  of  more  than  three  terms,  as 


EXTRACTION  OF  ROOTS.  215 

a-\-y-\-z-\-x  .  .  .  .  p,  let,  as  before,  u=  the  sum  of  all  the  terms  after 
the  first.     Then,  a+w  will  be  equal  to  the  given  polynomial,  and 

From  which  we  see,  that  by  cubing  a  polynomial,  we  obtain  the 
cube  of  the  first  term,  plus  three  times  ike  square  of  the  first  term 
multiplied  hy  each  of  the  remaining  terms,  plus  other  terms. 

It  often  happens  that  u  contains  a,  as  in  the  polynomial  a^-\-ax-\-b, 
where  u=zax+i.  But  since  we  suppose  the  polynomial  arranged 
with  reference  to  a,  it  follows  that  a  will  have  a  less  exponent  in  u 
than  in  the  first  term. 

In  this  case  also,  the  co-efficient  of  u,  multiplied  by  the  first  term 
of  u,  will  be  irreducible  with  the  remaining  terms  of  the  develop, 
ment,  because  that  product  will  mvolve  a  to  a  higher  power  than 
the  other  terms :  and  when  a  does  not  enter  u,  the  product  of  that 
co-efficient  by  all  the  terms  of  w,  will  be  irreducible  with  all  the 
other  terms  of  the  development. 

185.  As  to  the  extraction  of  roots  of  polynomials,  we  will  first 
explain  the  method  for  the  cube  root ;  it  will  afterwards  be  easy  to 
generalize. 

Let  N  be  the  polynomial,  and  R  its  cube  root.  Conceive  the  two 
polynomials  to  be  arranged  with  reference  to  some  letter,  a,  for  ex- 
ample.  It  results  from  the  law  of  composition  of  the  cube  of  a  po- 
lynomial (Art.  184),  that  the  cube  of  R  contains  two  parts,  which 
cannot  be  reduced  with  the  others ;  these  are,  the  cube  of  the  first 
term,  and  three  times  the  square  of  the  first  term  by  the  second. 

Hence,  the  cube  root  of  that  term  of  N  which  contains  a,  affect- 
ed with  the  highest  exponent,  will  be  the  first  term  of  R  :  and  the 
second  term  of  jR  will  be  found  by  dividing  the  second  term  of  N 
by  three  times  the  square  of  the  first  term  of  R. 

If  then,  we  form  the  cube  of  the  two  terms  of  the  root  already 
found,  and  subtract  it  from  N,  and  divide  the  first  term  of  the  re- 
mainder by  3  times  the  square  of  the  first  term  of  R,  the  quotient 
will  be  the  third  term  of  the  root.  Therefore,  having  arranged 
the  terms  of  N,  we  have  the  following 


216  ALGEBRA. 

RULE. 

I.  Extract  the  cube  root  of  the  first  term. 

II.  Divide  the  second  term  of  N  by  three  times  the  sqiuire  of  the 
first  term  of  R  :  the  quotient  will  be  the  second  term  of  R. 

III.  Having  found  the  two  first  terms  of  R,form  the  cube  of  the 
binomial  and  subtract  it  from  N  ;  after  which,  divide  the  first  term  of 
the  remainder  by  three  times  the  square  of  the  first  terin  of  R  :  the 
quotient  will  be  the  third  term  of  R. 

IV.  Cube  the  three  terms  of  the  root  found,  and  subtract  the  cube 
from  N ;  then  divide  the  first  term  of  the  remainder  by  the  divisor 

already  used  :  the  quotient  will  be  the  fourth  term  of  the  root,  and  the 
remaining  terms,  if  there  are  any,  may  be  found  in  a  similar  manner. 

EXAMPLES. 

1.  Extract  the  cube  root  of  a;«— Gx^+lSx*  — 20xHl5a:*  — Gx  +  1. 

(sr'-2xf=x^-ex^  +  12x*-   Sa;^  3^^ 

1st  Rem.     .     ~.     '.     '.       3.r*— 12x3+,  &c. 
(r'-2x+lf=x''-6x'  +  15a'-20x^-{-15x^—6x+l. 

In  this  example,  we  first  extract  the  cube  root  of  x',  which  gives 
x^,  for  the  first  term  of  the  root.  Squaring  a^,  and  multiplying  by 
3,  we  obtain  the  divisor  3a;'* :  this  is  contained  in  the  second  term 
—  6x^,  —2x  times.  Then  cubing  the  root,  and  subtracting,  we  find 
that  the  first  term  of  the  remainder  3x*,  contains  the  divisor  once. 
Cubing  the  whole  root,  we  find  the  cube  equal  to  the  given  polyno- 
mial. 

Remark.  The  rule  for  the  extraction  of  the  cube  root  is  easily 
extended  to  a  root  with  a  higher  index.     For, 

Let  a+i-j-c-j-  .  .    f,  be  any  polynomial. 

Let  s=   the  sum  of  all  the  terms  after  the  first. 

Then  a  +  s=   the  given  polynomial :  and 

(a+5)"  =  a"  +  na''  *5+  other  terms- 


CALCULUS  OF  RADICALS.  217 

That  is,  the  n""  poiver  of  a  polynomial,  is  equal  to  the  n"'  pouter  of 
the  first  ter7n,  plus  n  times  the  first  term  raised  to  the  power  n— 1, 
multiplied  hy  each  of  the  remaining  terms ;  +  other  terms  of  the  de- 
velopment. 

Hence,  we  see,  that  the  rule  for  the  cube  root  will  become  the 
rule  for  the  n""  root,  by  first  extracting  the  n'^  root  of  the  first  term, 
taking  for  a  divisor  n  times  this  root  raised  to  the  n^l  power,  and 
raising  the  partial  roots  to  the  ?i"'  power,  instead  of  to  the  cube. 

2.  Extract  the  4th  root  of 

lQa*-QQa''x^2\Qa'x'-2\Gax'  +  Qla\ 

(2a-3a;)''  =  16a''-96a^x'+216aV-216ca;='+81a''l32a3=4x(2a)3 
We  first  extract  the  4th  root  of  16a*,  which  is  2a.  We  then 
raise  2a  to  the  third  power,  and  multiply  by  4,  the  index  of  the  root : 
this  gives  the  divisor  32a^.  This  divisor  is  contained  in  the  second 
term  —  96a^a;,  —  3a;  times,  which  is  the  second  term  of  the  root. 
Raising  the  whole  root  to  the  4th  power,  we  find  the  power  equal  to 
the  given  polynomial. 

3.  Find  the  cube  root  of 

a;6  +  6a;5— 40a^  +  96a;  -  64. 

4.  Find  the  cube  root  of 

Ibx^-Qx+x^-Qx" -203^ +  10x^  +  1. 

5.  Find  the  5th  root  of 

320^*  -  SOx* + 80.r^  -  40a;2  + 1  Ox  - 1 . 

Calculus  of  Radicals. 

186.  When  it  is  required  to  extract  a  certain  root  of  a  monomial 
or  polynomial  which  is  not  a  perfect  power,  it  can  only  be  indicated 
by  writing  the  proposed  quantity  after  the  sign  V,  and  placing  over 
this  sign  the  number  which  denotes  the  degree  of  the  root  to  be  ex- 
tracted. This  number  is  called  the  index  of  the  root,  or  of  the  radical. 

A  radical  expression  may  be  reduced  to  its  simplest  terms,  by 
19 


218  ALGEBRA. 

observing  that,  the  n""  root  of  a  product  is  equal  to  the  product  of  the 
n^^  roots  of  its  different  factors. 

Or,  in  algebraic  terms  : 

Vabcd=:i/aXV^X  V^X  V^. 

For,  raising  both  members  to  the  n""  power,  we  have  for  the  first, 
(  V  abed)  ^ahcd  .  .  .,     and  for  the  second, 
(V«X7&X  Vcx7^---)"=(^«)'"-(V^)"-(Vc)''.(yrf)"...=a^c<i. 

Therefore,  since  the  n"'  powers  of  these  quantities  are  equal,  the 
quantities  themselves  must  be  equal. 

Let  us  take  the  expi-ession  V54a*i^c^  which  cannot  be  replaced 
by  a  rational  monomial,  since  54  is  not  a  perfect  cube,  and  the  ex- 
ponents of  a  and  c  are  not  divisible  by  3  :  but  we  can  put  it  under 
the  form 


V54fl^6^c^=  V-ilaV.  V2ac^=3ab  V2ac^ 
In  like  manner, 

V8^=  2  VaF;    V48a^«  =:2aPc  VSac- ; 
Vl92a'bc'-=  V64aV-x  VSab=^2ac''  VSab. 

In  the  expressions,  Sab  'V2a?,  2  Va%  2ah^c  VSac^,  the  quanti- 
ties placed  before  the  radical,  are  called  co-efficients  of  the  radical. 

187.  The  rule  of  (Art.  125)  gives  rise  to  another  kind  of  simpli- 
fication. 

Take,  for  example,  the  radical  expression,  V4a^  ;  from  this  rule 

we  have,  V4a^=\/  Vio^  and  as  the  quantity  affected  with  the 
radical  of  the  second  degree  v/,  is  a  perfect  square,  its  root  can  be 
extracted,  hence 

V4^=  \^. 
In  like  manner, 


Ja^'/r*  ^^ 


V^Qa'lr'  ='V     y36«^6''  = 


CALCULUS  OF  RADICALS.  219 

In  general,  Va"="'v/^a"=:  Va  ;  that  is,  when  the  index  of  a 
radical  is  multiplied  by  any  number  ii,  and  the  quantity  under  the 
the  radical  sign  is  an  exact  n"'  power,  ive  can,  without  changing  the 
value  of  the  radical,  divide  its  index  by  n,  and  extract  the  n""  root  of 
the  quantity  under  the  sign. 

This  proposition  is  the  inverse  of  another,  not  less  important,  viz. 
ive  can  multiply  the  index  of  a  radical  Iry  any  number,  provided  we 
raise  the  quantity  under  the  sign  to  a  power  of  which  this  number 
denotes  the  degree. 

Thus,  Va^^'Va".     For,  a  is  the  same  thing  as   'Va";  hence, 
y  a—  V  ^a"=z  Va". 

This  last  principle  serves  to  reduce  two  or  more  radicals  to  the 
same  index. 

For  example,  let  it  be  required  to  reduce  the  two  radicals  v  2a 
and    V{a+b)  to  the  same  index. 

By  multiplying  the  index  of  the  first  by  4,  the  index  of  the  se- 
cond, and  raising  the  quantity  2a  to  the  fourth  power  ;  then  multi- 
plying the  index  of  the  second  by  3,  the  index  of  the  first,  and 
cubing  a+b,  the  values  of  the  radicals  will  not  be  changed,  and 
the  expressions  will  become 

V2^='V'2V='VT6^;    V{a+b)='V(a+by. 
188.  Hence  to  reduce  radicals  to  a  common  index  we  have  the 

following 

RULE. 

Multiply  the  index  of  each  radical  by  the  product  of  the  indices  of 
all  the  other  radicals,  and  raise  the  quantity  under  each  radical  sign 
to  a  power  denoted  by  this  product. 

This  rule,  which  is  analogous  to  that  given  for  the  reduction  of 
fractions  to  a  common  denominator,  is  susceptible  of  some  modifi- 
cations. 


220 


ALGEBRA. 


For  example,  reduce  the  radicals  W a,  Wbh,  Wa--\-P,  to  the 
same  index. 

As  the  numbers  4,  6,  8,  have  common  factors,  and  24  is  the  most 
simple  multiple  of  the  three  numbers,  it  is  only  necessary  to  multi- 
ply the  first  by  6,  the  second  by  4,  and  the  third  by  3,  and  to  raise 
the  quantities  under  each  radical  sign  to  the  6th,  4th,  and  3d  pow- 
ers respectively,  which  gives 

In  applying  the  above  rules  to  numerical  examples,  beginners 
very  often  make  mistakes  similar  to  the  following,  viz.  :  In  reduc- 
ing the  radicals  \/2  and  -v/3  to  a  common  index,  after  having  mul- 
tiplied  the  index  of  the  first  (3),  by  that  of  the  second  (2),  and  the 
index  of  the  second  by  that  of  the  first,  then,  instead  of  multiplying 
the  exponent  of  the  quantity  under  the  first  sign  by  2,  and  the  expo- 
nent of  that  under  the  second  by  3,  they  often  multiply  the  quantity 
under  the  first  sign  by  2,  and  the  quantity  under  the  second  by  3. 
Thus,  they  would  have 

'->/¥= ''-v/2^='^/l;     and     V3=V3^=:V'9: 
Whereas,  they  should  have,  by  the  foregoing  rule, 

''^r2=W~{2f='^r^,    and     V^  =  '\^J^=V^. 
Reduce  V2,  %/4,  W\,  to  the  same  index. 

Addition  and  Subtraction  of  Radicals. 
189.  Two  radicals  are  similar,  when  they  have  the  same  index, 
and  the  same  quantity,  under  the  sign.     Thus,  3  -^/ab  and  7  Vah, 
are  similar  radicals,  as  also  3a-  V^^j  and  9c^  %/P. 

Therefore,  to  add  or  subtract  similar  radicals,  add  or  subtract 
their  co-efficients,  and  prefix  the  sum  or  difference  to  the  common 
radical. 

Thus,     2,Wb  +  )lWb=bWb,    ^Wb-2Wb=Wb, 
3aVbdz2cVb=(3adz2c)Vb. 


CALCULUS  OF  RADICALS.  221 

Sometimes  when  two  radicals  are  dissimilar,  they  can  be  reduced 
to  similar  radicals  by  Arts.  186  and  187.     For  example, 

W8a'b+l6a''-Vb'  +  2aP=2aVb  +  2^-b  VM^«~ 

=.{2a-b)Vb+2^; 
3  V4^+2  V2^=3  V2^+2  V2a=5  V2^. 
When  the  radicals  are  dissimilar,  and  irreducible,  they  can  only 
be  added  or  subtracted  by  means  of  the  signs  +  or  — . 

Multiplication  and  Division. 
190.  We  will  first  suppose  that  the  radicals  have  a  common 
index. 

Let  it  be  required  to  multiply  or  divide  Va  by  V^.    We  have 

Vax  Vb=  Vab,  and    Va -^   Vb=\/-r. 

For  by  raising  Va  .  "Vb  and  'Vab  to  the  n""  power,  we  obtain 

the  same  result  ab ;  hence  the  two  expressions  are  equal. 

r/ffl        1     "  /^       .     ,       ,        .  .       « 

In  Uke  manner,  — r  and   \/  -  raised  to  the  n'''  power  give  -^: 
iy  0  ^     0  .  0 

hence  these  two  expressions  are  equal.     Therefore  we  have  the 

following 

RULE. 

Multiply  or  divide  the  quantities  tinder  the  sign  by  each  other,  and 
give  to  the  product,  or  quotient,  the  common  radical  sign.  If  they 
have  co-efficients,  first  multiply  or  divide  them  separately. 

Thus, 

2a V    _Z_x-3aV  ^     T     ^  =-6a^V  ^ j^-. 

c  a  ca 

or,  reducing  to  its  simplest  terms, 

Qa^a^  +  V) 

19* 


222  ALGEBRA. 

3fl  VSafx^i  V4aF^=6ai  VS2a*c=l2a''bV2'c 


V 


\/^ 


aW+b'  _  '  /Qb{aW  +  h^)  _^^l  /g!±jl 


Sb 

When  the  radicals  have  not  a  common  index,  they  should  be  re- 
duced  to  one. 

For  example,      3aVbx5b''V^c=ldabx'Vsb^' 

EXAJtfPLES, 


1.  Multiply    \/2X  -V3    by  V  yX' 

^-w 

Alls.    'ViT 

2.  Multiply  2  Vis  by  sVTo 

An*.     6V337500. 

5  /T      .  /y 

3.  Multiply  4V  Y  by  2  V  -j- 

':  /  27 

^--  ^v  ,5,. 

2  'v/  3  V  -v/  4 

4.  Reduce =  to  its  lowest  terms. 

x*-v/TxV3 


^715.     4'V288. 


/V  J-  X2V  3 

5.  Reduce   \/ ^ ^z  to  its  lowest  terms. 

^  4V2  X  -/s 

6.  Multiply  VT,  VT,  and  'V^  to  ether. 

An*.      'V648000. 


CALCULUS  OP  RADICALS.  223 


7.  Multiply    V    -r-,    'v    17  and   'V6    together. 


3'      "      2 


Ans.      V  ^. 


8.  Multiply    (4V-^+5\Ai)  by    (V  Y+2V  y) 

9.  Divide    yV  Y  by   (  -/T+sVy) 


43     13     ^- 

y +Y  v42. 


^n..     1 


10.  Divide  1  by  *VT+V3  . 

*V^-  *V~^+ W^-  V"^ 


Ans. 


a—b 
11.  Divide  *V~a"  +  *-v/X  by  W  a —*V1)' . 

a-\-h-\-2  V^+2Va^+2*Va^ 


-4ns. 


a-b 


Formation  of  Powers,  and  Extraction  of  Roots. 
191.  By  raising  V«  to  the  n""  power,  we  have 

(Va)''=V«XV«XV«  •  •  •  ='Va", 
by  the  rule  just  given  for  the  multiplication  of  radicals.     Hence, 
for  raising  a  radical  to  any  power,  we  have  the  following 

RULE. 

Raise  the  quantity  under  the  sign  to  the  given  power,  and  affect 
the  result  with  the  radical  sign,  having  the  primitive  index.  If  it 
has  a  co.efficient,  first  raise  it  to  the  given  pozver. 

Thus,         ( *V^f = V{^ay = Vl6a«= 2a  V^; 

(3V2^)5=35.V(2^'=243V32^5=:486aV4^ 


224  ALGEBRA. 

When  the  index  of  the  radical  is  a  multiple  of  the  power,  the  re- 
sult can  be  reduced. 

For,  *-\/2a='\/  V2a  (Art.  176)  :  hence,  to  square  *V2a,  we 
have  only  to  omit  the  first  radical,  which  gives  (  V2a)  =  V2a. 

Again,  to  square  Wsb,  we  have   V  3b=\/  VsZ*  •  hence 

Consequently,  when  the  itidex  of  the  radical  is  divisible  by  the  ex- 
ponent of  the  power,  perform  this  division,  leaving  the  quantity  under 
the  radical  unchanged. 

To  extract  the  root  of  a  radical,  multiply  the  index  of  the  radical 
by  the  index  of  the  root  to  be  extracted,  leaving  the  quantity  under  the 
sign  unchanged. 

Thus,         V   *V3^="V^;   S/^VTc^WVc. 

This  rule  is  nothing  more  than  the  principle  of  Art.  176,  enun- 
dated  in  an  inverse  order. 

When  the  quantity  under  the  radical  is  a  perfect  power,  of  the 
degree  of  either  of  the  roots  to  be  extracted,  the  result  can  be  re- 
duced. 

Thus,  \/  VSa^  bemg  equal  to  \/  VSo^  it  reduces  to  V2a. 

In  like  manner,  \/  W^^s/  Vdc^^^VSa. 
It  is  evident  that   y'l^a—  V  ya  ;  because  both  expressions  are 
equal  to  "'ya~(Art.  176). 

192.  The  rules  just  demonstrated  for  the  calculus  of  radicals, 
principally  depend  upon  the  fact  that  the  ji"*  root  of  the  product  of 
several  factors  is  equal  to  the  product  of  the  n""  roots  of  these  fac- 
tors ;  and  the  demonstration  of  this  principle  depends  upon  this  : 
When  the  powers,  of  the  same  degree,  of  two  expressions  are  equal, 


CALCULUS  OF  RADICALS.  225 

the  expressions  are  also  equal.  Now  this  last  proposition,  which  is 
true  for  absolute  numbers,  is  not  always  true  for  algebraic  expres- 
sions. 

To  prove  this,  we  will  show  that  the  same  number  can  have 
more  titan  one  square  root,  cuie  root,  fourth  root,  ^c. 

For,  denote  the  general  expression  of  the  square  root  of  a  by  x, 
and  the  arithmetical  value  of  it  by  ^  ;  we  have  the  equation  x^=a, 
or  xi^z=p^,  whence  x=zkp.  Hence  we  see  that  the  square  of  p, 
which  is  the  root  of  a,  will  give  a,  whether  its  sign  be  +  or  — . 

In  the  second  place,  let  x  be  the  general  expression  of  the  cube 
root  of  a,  and  p  the  numerical  value  of  this  root ;  we  have  the 
equation 

x^=a,  or  x^^p^. 

This  equation  is  satisfied  by  making  x=p. 

Observing  that  the  equation  cc^=^p^  can  be  put  under  the  form 
sP—p^=0,  and  that  the  expression  x^—p^  is  divisible  by  x—p,  (Art. 
59),  which  gives  the  exact  quotient,  x^-^-px+p^,  the  above  equation 
can  be  transformed  into 

(x—p)  (x^ -{-px-{-p^)^0. 

Now,  every  value  of  x  which  will  satisfy  this  equation  will  satis- 
fy  the  first  equation.  But  this  equation  can  be  verified  by  suppos- 
ing x—p^O,  whence  x=p  ;  or  by  supposuig 

x^+px+p'^—O, 
from  which  last  we  have 

x=--±-V-3,  or  x:=p[ j. 

Hence,  the  cube  foot  of  a,  admits  of  three  different  algebraic  va- 
lues, viz. 

P>    P{ 2 J'  ^""^  P\ 2 /• 

Again,  resolve  the  equation  x'^^p*,  in  which  p  denotes  the  arith- 
metical value  of  \/a.  This  equation  can  be  put  under  the  form 
x*— p''=0.    Now  this  expression  reduces  to  (a^— p^)  (a^+P^)* 


226  ALGEBRA. 

Hence  the  equation  reduces  to  (a^— p^)  {xr'-\-p^)=0,  and  can 
be  satisfied  by  supposing  x^—p^—0,  whence  x=±p  ;  or  by  suppos- 
ing a^-{-p-=0,  whence  x=±  V  —p'^—dzp  V—1. 

We  therefore  obtain  four  different  algebraic  expressions  for  the 
fourth  root  of  a. 

For  another  example,  resolve  the  equation  ....  x^z=p^, 
which  can  be  put  under  the  form :x^—p^=0. 

Nowa;®—2>^  reduces  to {s(P—p^)  (aP-\-p^), 

therefore  the  equation  becomes     ....    (x^—p^)  {xP+p^)=0. 

But  x^—p^—0,  gives 

/-1±  V~^3. 
x=p,  and  x=pl ; I. 

And  if  in  the  equation  x^+p^  =  0,  we  make  p~—p',  it  becomes 
aP—p'^—0  from  which  we  deduce  a;=|)',  and 

-=P( ^ ); 

or,  subslituthig  for  p'  its  value,  —p, 

/-1±  a/^ 
x=—p  and  x——pl J. 

Therefore  the  value  of  x,  in  the  equation  a;"— ^"=0,  and  conse- 
quently  the  6th  root  of  a,  admits  of  six  values,  p,  ap,  a'p,  —p, 
—  a]),  —a'p,  by  making 

-= 2- ,     a'= . 

We  may  then  conclude  from  analogy,  that  x  in  every  equation  of 
the  form  x'^—a—O,  or  a^'"— ^"=0,  is  susceptible  of  mdifferent  va- 
lues,  that  is,  the  m""  root  of  a  number  admits  of  m  different  alge. 
braic  values. 

193.  If  in  the  preceding  equations  and  the  results  corresponding 
to  them,  we  suppose  as  a  particular  case  a=l,  whence  p=l,  we 
shall  obtain  the  second,  third,  fourth,  &c.  roots  of  unity.  Thus 
+  1  and  —1  are  the  two  square  roots  of  unity,  because  the  equation 
ar*— 1  =  0,  gives  a:=zhl. 


CALCrLTJS  OF  RADICALS.  227 

^    ,.,  -l+V-3     -1—  VT 

In  like  manner   +1, , ,    are  the  three 

cube  roots  of  unity,  or  the  roots  of  a;^— 1=0.         And 

+  1,-1,  +  V  —  1,  —  V  — 1,  are  the  four  fourth  roots  of  unity, 
or  the  roots  of  x*— 1  =  0, 

194.  It  results  from  the  preceding  analysis,  that  the  rules  for  the 
calculus  of  radicals,  which  are  exact  when  applied  to  absolute  num- 
bers, are  susceptible  of  some  modifications,  when  applied  to  expres- 
sions or  symbols  which  are  purely  algeiraic ;  these  modifications  are 
more  particularly  necessary  when  applied  to  imaginary  expressions, 
and  are  a  consequence  of  what  has  been  said  in  (Art.  192). 

For  example,  the  product  of  V  —ahy  V  —a,  by  the  rule  of 
(Art.  190),  would  be 

Now,  Va"  is  equal  to  ±a  (Art.  192)  ;  there  is,  then,  apparent- 
ly,  an  uncertainty  as  to  the  sign  with  which  a  should  be  affected. 
Nevertheless,  the  true  answer  is  —a ;  for,  in  order  to  square  -y/m, 
it  is  only  necessary  to  suppress  the  radical ;  but  the  V  — «  X  V'  —  a 
reduces  to  (  V  — «}  ,  and  is  therefore  equal  to  —a. 

Again,  let  it  be  required  to  form  the  product  V  —a  x  v/  —h, 
by  the  rule  of  (Art.  190),  we  shall  have 

V  —a  X  V—b—  V+ab. 
Now,    Vab=±p  (Art.  192),  p  being  the  arithmetical  value  of 
the  square  root  of  ah ;  but  I  say  that  the  true  result  should  be  —p 
or  —  ■yob,  so  long  as  both  the  radicals   V —a  and   V—b  are  con. 
sidered  to  be  affected  with  the  sign  +. 

For,        V—a—  y/a.  V^l  and   V^F^  y/b.  V^ ; 
hence 

-/^x  V^=  va.  V-ix  V^-bx  V^r=  V~^h{  V^lf 

=  Vabx—l=—  Vab. 


228 


Upon  this  principle  we  find  the  different  powers  of  v— 1  to  be, 
as  follows  : 

V-i=  V^-i^  {V^^Y=-i, 

and       (a/3I)4^(  V^)^(-v/^)2=-lX-l=  +  l• 
Again,  let  it  be  proposed  to  determine  the  product  of   V  —  a  by 
the   V —h  which,  from  the  rule,  will  be   V +ah,  and  consequently 
will  give  the  four  values  (Art.  192). 

+  Vo^, . - Wab,  +  Wab.  V -I,  —  Wab.  V^T. 
To  determine  the  true  product,  observe  that 

But    V^^Tx *V^i^{V^y=  (v  V~^J  =  V^T^ 

hence  V  — a  .*V^^=iWab.  V—1. 

We  will  apply  the  preceding  calculus  to  the  verification  of  the 

expression ,    considered  as    a  root  of  the  equation 

a;3_l  =  0,  that  is,  as  the  cube  root  of  1  (Art,  192). 
From  the  formula         {a+by=a''  +  ^a''b  +  Sab''+P, 


we  have 


(_i)3+3(-i)^  V-3+3(-i).(  V^y+i  V-ny 

8 
—  I+SV^— 3x— 3  — 3  V^-3 


=  1. 


i_  V-s 


The  second  value, 
manner. 


may  be  verified  in  the  same 


Theory  of  Exponents. 

195.  In  extracting  the  n"'  root  of  a  quantity  a",  we  have  seen 
that  when  m  is  a  multiple  of  n,  we  should  divide  the  exponent  m  by 


THEORY  OF  exponents; .  229 

11  the  index  uf  the  ruuL  j  but  whon  tti  Is  not  divisible  by  n,  in  which 
case  the  root  cannot  be  extracted  algebraically,  it  has  been  agreed 
to  indicate  this  operation  by  indicating  the  division  of  the  two  ex. 
ponents. 

Hence,  "V  a"'=a~,  from  a  convention  founded  upon  the  rule  for 
the  exponents,  in  tlie  extraction  of  the  roots  of  monomials.  In  such 
expressions,  the  numerator  indicates  the  power  to  which  the  quantity 
is  to  he  raised,  and  the  denominator,  the  root  to  be  extracted. 

2  7 

Therefore,  ya^'z^a^  ;     V«'— «*• 

In  like  manner,  suppose  it  is  required  to  divide  a""  by  a".  We 
know  that  the  exponent  of  the  divisor  should  be  subtracted  from  the 

a'" 
exponent  of  the  dividend,  when  m>w,  which   gives     —z=a"'  ". 

But  when  ?ra<n,  in  wliich  case  the  division  cannot  be  effected  alge- 
braically, it  has  been  agreed  to  subtract  the  exponent  of  the  divisor 
from  that  of  the  dividend.  Let  p  be  the  absolute  difference  between 

n  and  m  ;  then  will  n^m+p,  whence  -^^=a-?  ;  but  -^p 
reduces  to     — ;     hence     «^=^- 

Therefore,  the  expression  a-P  is  the  symbol  of  a  division  which  it 
has  been  impossible  to  perform  ;  and  its  true  value  is  the  quotient 
represented  by  unity  divided  by  the  letter  a,  affected  with  the  ex- 
ponent p,  taken  positively.     Thus, 

a-^  a^ 

The  notation  of  fractional  exponents  has  the  advantage  of  giving 
an  entire  form  to  fractional  expressions. 

From  the  combination  of  the  extraction  of  a  root,  and  an  impos- 
sible division,  there  results  another  notation,  viz.  negative  fractional 
esvonents. 

20 


230  ALGEBRA. 

In  extracting  the  /*'*  root  of    — ,  we  have  first  — =a~",  hence 
a"  a"" 

"    /   1  n    / _™ 

\/  —=  Va  "'=a  " ,  by  substituting  a  fractional  exponent  for 
the  radical  sign. 

Hence,  a",  wp,  a  ",  are  conventional  expressions,  founded  wp. 
on  preceding  rules,  and  equivalent  to  "Va"",  — ,    \/  — . 

We  may  therefore  substitute  the  second  for  the  first,  or  recipro- 
cally. 

As  aP  is  called  a  to  the  p  power,  when  ^  is  a  positive  whole  num- 
ber, so  by  analogy,  a~,  a-^,  a~  %  are  called  a  to  the  —  power,  a  to 

n   ^ 

the  -p  power,  a  to  the   — —  power,  which  has  induced  algebra- 

ists  to  generalize  the  ^woIA power;  but  it  would,  perhaps,  be  more 

accurate  to  say,  a,  exponent  ~,  exponent   -p,   exponent   -— 

n 
using  the  word  power  only  when  we  wish  to  designate  the  product 
of  a  number  multiplied  by  itself  two  or  more  times. 

1  1 

Smce  a~P  and  —  are  equivalent  expressions,  also  a^  and  . 

^  a~''  ' 

we  conclude  that  any  factor  may  he  transferred  from  the  numerator 
to  the  denominator,  or  from  the  denominator  to  the  numerator,  hy 
changing  the  sign  of  its  exponent. 

Multiplication  of  Quantities  affected  with  any  Exponents. 

3  2 

196.  In  order  to  multiply  a^  by  a^,  it  is  only  necessary  to  add 
the  two  exponents,  and  we  have 

3  2.  3.2  ia. 


For,  by  (Art.  195),  a*  =  «/«'  ;  a^  =  V«^ 


3  2 

hence.  a^  va^  —  s/flSv  3/«2 


THEORY  OF  EXPONENTS.  231 

or,  performing  the  multiplication  by  the  rule  of  (Art.  190), 
3       2  la 


a  *  =  ^^,  J=Va'; 


Again,  multiplying  a  *  by  a® ,  we  have 

for, 
Hghcg 

"-*xa*=yix  v^=  V;^x  ■v;r.= V^"=  'vr=<.^^ 

In  general,  multiplying  a   "    by  a '  ;  we  have 

a   "  X«'  =a   "    *  =a  "»    . 
Therefore,  in  order  to  multiply  two  monomials  affected  with  any 
exponents  whatever,  add  together  the  exponents  of  the  same  letter ; 
this  rule  is  the  same  as  that  given  in  (Art.  41),  for  quantities  affect- 
ed with  entire  exponents. 

From  this  rule  we  will  find  that 

a  -JL  2  ^       i_i  J.  _2 

a*b  ^c-^xa'b^c'^a^  b'c  ^  ; 

and  3a-2#x2a'^^>^c2=6a"'^*5«c^ 

Division. 

197.  To  divide  one  monomial  by  another  when  both  are  affected 
with  any  exponent  whatever,  follow  the  rule  given  in  Art.  50  for 
quantities  affected  with  entire  and  positive  exponents ;  that  is,  sub. 
tract  the  exponents  of  the  letters  in  the  divisor  from  the  exponents  oj 
the  same  letters  in  the  dividend. 

For,  the  exponent  of  each  letter  in  the  quotient  must  be  such, 
that  added  to  that  of  the  same  letter  in  the  divisor,  the  sum  shall 
be  equal  to  the  exponent  of  the  letter  the  dividend ;  hence  the  ex- 
ponent in  the  quotient  is  equal  to  the  difference  between  the  expo- 
nent  in  the  dividend  and  that  in  the  divisor 


232  ALGEBRA. 

EXAMPLES. 

3  i.  3_4  __1_ 

/7*_l./75— «*        5—^      20    . 


2  3  _1     7  _9_    _ 


Formation  of  Powers. 
198.  To  form  the  n"'  power  of  a  monomial,  affected  with  any- 
exponent  whatever,  observe  the  rule  given  in  Art.  182,  viz.  multi. 
ply  the  exponent  of  each  letter  by  the  exponent  m  of  the  power  ;  for, 
to  raise  a  quantity  to  the  m"'  power,  is  the  same  thing  as  to  multi- 
ply  it  by  itself  m  —  1  times;  therefore,  by  the  rule  for  multiplica- 
tion,  the  exponent  of  each  letter  must  be  added  to  itself  m—1  times, 
or  multiplied  by  ?n< 

^  /  sXs        15/2X3        6 

Thus,     \a*}   =a*  ;   {a"")   =a^  =  a^  ; 

(-XsXe  -0-9         /_s\i2  _10 

2a    ^b*)   =64a    '^^  .    [^^    <^)      =a        . 

Extraction  of  Roots. 

189.  To  extract  the  n""  root  of  a  monomial,  follow  the  rule  given 
in  Art.  182,  viz.  divide  the  exponent  of  each  letter  by  the  index  of 
the  root. 

For,  the  exponent  of  each  letter  in  the  result  should  be  such, 
that  multiplied  by  n,  the  index  of  the  root  to  be  extracted,  there 
will  be  produced  the  exponent  with  which  the  letter  is  affected  in 
the  proposed  monomial ;  therefore,  the  exponents  in  the  result  must 
be  respectively  equal  to  the  quotients  arising  from  the  division  of 
the  exponents  in  the  proposed  monomial,  by  n,  the  index  of  the 
root. 


Thus, 


V. 


3    _a         1-3 
a'b      =a'b   ^ 


THEORY  OF  EXPONENTS. 


233 


The  last  three  rules  have  been  easily  deduced  from  the  rule  for 
multiplication ;  but  we  might  give  a  direct  demonstration  for  them, 
by  going  back  to  the  origin  of  quantities  affected  with  fractional 
and  negative  exponents. 

We  will  terminate  this  subject  by  an  operation  which  contains 
implicitly  the  demonstration  of  the  two  preceding  rules. 

Let  it  be  required  to  raise  a"   to  the  — —  power; 
We  say  ihat, 

For,  by  going  back  to  the  origin  of  these  notations,  we  find  that 

The  advantage  derived  from  the  use  of  exponents  consists  prin- 
cipally in  this  :  The  operations  performed  upon  expressions  of  this 
kind  require  no  other  rules  than  those  established  for  the  calculus 
of  quantities  affected  with  entire  exponents.  Besides,  this  calculus 
is  reduced  to  simple  operations  upon  fractions,  with  which  we  are 
already  familiar. 

200.  Remark.  In  the  resolution  of  certain  questions,  we  shall 
be  led  to  consider  quantities  affected  with  incommensurable  expo, 
nents.  Now,  it  would  seem  that  the  rules  just  established  for  com- 
mensurable  exponents,  ought  to  be  demonstrated  for  the  case  in 
which  the  exponents  are  incommensurable ;  but  we  will  observe, 
that  an  incommensurable,  such  as  V  3  ,  Vll,  is  by  its  nature  com- 
posed  of  an  entire  part,  and  a  fraction  which  cannot  be  expressed 
exactly,  but  to  which  it  is  possible  to  approximate  as  near  as  we 
please,  so  that  we  may  always  conceive  the  incommensurable  to  be 
replaced  by  an  exact  fraction,  which  only  differs  from  it  by  a  quan- 
20* 


234  ALGEBRA. 

tity  less  than  any  given  quantity ;  and  in  applying  the  rules  to  the 
symbol  which  designates  the  incommensurable,  it  is  necessary  to  un- 
derstand that  we  apply  it  to  the  exact  fraction  which  represents  it 
approximatively. 

EXAMPLES. 


iveduce — — to  its  simplest  terms. 

2  V  * 


\  2V2(3)2  J 


Ans.     4  V  3  . 


Reduce     ^  ^^  '    y     to  its  sii-nplest  terms 

:V2(3)' 

1 

Am 


384 


^A  ar+  V3i  \  ^ 

(    2v/2.(f)^    ) 


Reduce     \/  .<  ^-'     —  V     to  its  simplest  terms. 

Ans.     \/y(-^VT+V'2l). 

Demonsti'ation  of  the  Binomial  Theorem  in  the  case  of  any 
Exponent  whatever. 

201.  Since  the  rules  for  the  calculus  of  entire  and  positive  expo- 
nents  may  be  extended  to  the  case  of  any  exponent  whatever,  it  is 
natural  to  suppose  that  the  binomial  formula,  which  serves  to  deve- 
lop  the  m"'  power  of  a  binomial  when  m  is  entire  and  positive,  will 
also  effect  this  when  m  is  any  exponent  whatever.  In  fact,  analysts 
have  discovered  that  this  is  the  case,  and  they  have  deduced  im- 
portant consequences  from  it,  both  for  the  extraction  of  roots  by  dp- 
j)roximation,  and  the  development  of  algebraic  expressions  into  series. 

The  following  is  a  modification  of  Euler's  demonstration. 

We  will  remark,  in  the  first  place,  that  the  bmomial  x+a  can 

be  put  under  the  form  x(l-\ — )  ;  whence  there  results 


BINOMIAL  THEOREM.  235 

{x+ay"=x-^(l+-^^  =a;'"(l+2)%  by  making  —=z. 

Therefore,  if  the  formula 

m  — 1     „  711— I        m  —  2 

{l+2)"'=l+m2+m — - — s?-\-m. — ~ — .  — g— s^+,&c.  (A) 

is  proved  to  be  correct  for  any  value  of  ?/?,  we  may  consider  the 
formula. 

711—1 

(x-\-a)'"=x'"+7nax^~^+vi .  — — — a^x'^~^ 

7)1  —  1  7)1  —  2 

+m .  — -— .  — ^—a'x^-'+,  &c.  (B) 


exact  for  any  value  of  m.     For,  by  substituting    —    for   z   in  the 
formula  (A)^  and  multiplying  by  a;",  we  obtain 

(x+ay=x'"{l+m—+7)i  .—J— .—+,  &c.], 

from  which,  by  performing  the  operations  mdicated,  we  obtain  the 
formula  (B). 

Now,  when  m  is  a  whole  number,  we  have 

m— 1    „  7)1—1       7)1—2 

(l+zf^l+mz  +  m. Z^  +  7)l. — -—.  — ^—2'  +  ,  &c. 

P 
But,  if  m  is  a  fraction    — ,     we  do  not  know  from  what  algebraic 

expression  the  development 

7)1-1  7)1—1  7)1  —  2 

l+7)iz+7)i — — — z-+7)i. — - — . — - — 2^.^,  &c.  ...  is  derived. 
Denoting  this  unknown  expression  by  y,  we  have  the  equation 

7)1-1      „  TO— 1  7)1  —  2 

y==l+niz+7)i.—^—z-+))i.—^—.—^—r^+,  &c (1). 

and  it  is  now  required  to  prove  that  y=(l+z)"'. 

If  m'  is  another  fractional  exponent,  we  shall  have  in  like  manner, 

m'  — 1    „  7)i'  —  l      7)i'—2 

3/'=l+m'2+OT' . -^— z^+m' .-—— .  — — — r^+,&c. . .  (2). 


236  ALGEBRA. 

Multiplying  the  equations  (1)  and  (2),  member  by  member,  we 
shall  have  for  the  first  member  of  the  result  yy'.  As  to  the  second, 
it  would  be  very  difficult  to  obtain  its  true  form,  by  the  common 
rule  for  the  multiplication  of  polynomials ;  but  by  observing  that 
the  form  of  a  product  does  not  depend  upon  the  particular  values  of 
the  letters  which  enter  into  its  two  factors  (Art.  47),  we  see  that  the 
above  product  will  be  of  the  same  form  as  in  the  case  where  m  and 
m'  are  positive  whole  numbers.     Now  in  this  case  we  have 


\^mz+m  .— ^— z2+  .  .  .  ={l+zY, 


Ij^rn'z+m' . ^— z'+  •  •  •  ={l+zY', 


m—\ 

m!  —  \ 

whence 

l+mz+7n. — - — 2^+  .  .  .  j{\+m'z+m' . — - — z'+ j 

m+m'  —  \ 
=  {l+zY+^'=lJf.{m+m')z  +  {ra+m') z^+ ; 

Therefore  this  form  is  true  in  the  case  in  which  m  and  m'  are  any 

quantities  whatever,  and  we  have 

m+m'  —  1 
yy'=l  +  {m+m')z+{m-^m') ^ .  5r+ (3); 

Let  m"  be  a  third  positive  fractional  exponent,  we  shall  have 

m"  —  \ 
y'  =l+ni"z+m" ^— 5^^+  •  •  • 

Multiplying  the  two  last  equations  member  by  member,  we  have 

,  „            ,                                                 m-\-7n' +m"  —1 
yyy  =l-^{m+7n' +m")z-\-{m+m' +m") z-+ 

V 
Suppose  the  fractional  exponent  m= — .  Take  as  many  exponents 

m,  m',  m",  m'",  &c.  as  there  are  units  in  q  ;  we  shall  have,  by  mak- 
ing r  equal  to  the  sum  of  the  exponents  wi+m'+?«"+OT"'+  .  •  • 

yy  3/  y"=i +r2+/- .  -^^H^  •  —^  •  ~3~^+  •  •  •  (4)- 


BIN03IIAL  THEOREMS.  237 

And  by  supposing  m—m'=m"—7n"'  ...  in  which  case 
r=m+m+?n+??i+  .  .  .  =^m<i, 
the  equation  (4)  becomes 

mq  —  1  mq  —  1      mq  —  2 

y^=l+mq.z-{-mq. — z^+m^. -— . 3—2^+  •  •  • 

P 

Now  we  have  by  hypothesis,  m= — ,  or  mq=p  ; 

but  p  is  a  whole  number,  therefore  the  second  member  of  this  equa- 
tion is  the  development  of  (l+z)^,  which  gives  ?/'=(l4-z)'',  whence 

p_ 
t/=:(l+2)«  =(l+z)'" ;  consequently 


m  being  any  positive  fraction. 

To  demonstrate  this  formula,  for  the  case  in  which  m  is  a  negative 

fraction  or  whole  number,  it  is  only  necessary  to  suppose,  m'=^  —  m, 

in  the  equation  (3)  obtained   from  the  equations  (1)  and   (2),  for 

when  m+m'—Q,  the  equation   (3)   reduces   to  yy'=^\;    whence 

1 

But  since  m  is  negative  by  hypothesis,  m'  or  —m,  must  be  posi- 
tive, and  we  have 

y'={\+zY',  hence  y=-^^^^—^={\+z)^'={l+zY, 


and  consequently 

m—\ 
{\-\-zY=\+mz  +  m——-.z^+  ...     or 

-m'-l  ,        (_m'-l)(-m'-2) 
(1-f 2)-™'=!  — m's— ??t' z^—m 1    ^    ^ "*  +'  ^^' 


238 


Ajjplications  of  the  Binomial  Theorem. 

202.  If  in  the  formula 

/  a  m—\      a^  m  —  1 

1  i 

we   make     m— — ,     it  becomes     {x-\-ay 


{x+ar= 

m-2 

3 

1  l__    J_ 

lain  a^        1      n  n 


or,  reducing, 

2n— 1 


(1       a       1     71  —  1    a-       1     «— 1 
71  '  X       n  '    2n    '  sP       n  '    2n 


3?i        s? 

3?j- 


The  fifth  term  can  be  found  by  multiplying  the  fourth  by 


4n 


a 

and  by    — ,  then  changing  the  sign  of  the  result,  and  so  on. 

203.  Remark.  When  the  terms  of  a  series  go  on  decreasing  in 
value,  the  series  is  called  a  decreasing  or  converging  series ;  and 
when  they  go  on  increasing  in  value,  it  is  called  a  diverging  series. 

In  a  converging  series  the  greater  number  of  terms  we  take  in 
the  series,  the  nearer  will  we  approximate  to  the  true  value  of  the 
proposed  series.  When  the  terms  of  the  series  are  alternately 
positive  and  negative,  we  can,  by  taking  a  given  number  of  terms, 
determine  the  degree  of  approximation. 

For,  let  a  —  b  +  c  —  d-j-e—f-\-  .  .  .,  &c.  be,  a  decreasing  series  ; 
b,  c,d  .  .  .  being  positive  quantities,  and  let  x  denote  the  number 
represented  by  this  series. 

The  numerical  value  of  x  is  contained  bt^tween  any  two  consecu- 
tive sums  of  the  terms  of  the  series.  For  take  any  two  consecutive 
sums, 

a  —  h+c—d+e—f,  and  a—h+c—d+e—f-i-g. 


BliVOMIAL  THEOREM.  239 

In  the  first,  the  terms  which  follow     /,  are  g—h,  +  k—l4-  - 
but  silica  the  series  is  decreasing,  the  paTiial  differences  g—h, 

k—l are  positive  numbers  ;  therefore,  in  order  to  obtain  the 

complete  value  of  x,  a  certain  positive  number  must  be  added  to  the 
sum  a  —  h-\-c—d-{-e—f.     Hence  we  have 

a—h-{-c—d-\-e—f<:^x. 

In  the  second  series,  the  terms  which  follow  -\-g  are  —h-\-k, 
—l-\-m  ....  Now,  the  partial  differences  —h-\-k,  —l-\-ni  .  .  ., 
are  negative  ;  therefore,  in  order  to  obtain  the  sum  of  the  series,  a 
negative  quantity  must  be  added  to 

a—h+c—d+e—f+g, 
or,  in  other  words,  it  is  necessary  to  diminish  it.     Consequently 
a—h-{-c—d-[-e—f-{-gyx. 
Therefore,  x  is  comprehended  between  these  two  sums. 
The  difference  between  these  two  sums  is  equal  to  g.     But  since 
X  is  comprised  between  them,  their  difference  g  must  be  greater  than 
the  difference  between  x  and  either  of  them  ;  hence,  the  error  com- 
mitted  hy  taking  a  certain  number  of  terms,  a— b+c— d+e— f,  for 
the  value  of  x,  is  numerically  less  than  the  following  term. 

206.  The  binomial  formula  also  serves  to  develop  algebraic  ex- 
pressions into  series. 

Take  for  example,  the  expression ,  we  have 

1 

1  —  z       ^  ' 

In  the  binomial  formula,  make  7n=  — 1,  ic=l,  and  a=  —  z,  it  be. 
comes 

-1-1    -1-2 

-1.— -.— -.(-Z)3-... 


240  AI.OEBRA. 

or.  performing  the  operations,  and  ohsftrving  that  each  term  is  com- 
posed of  an  even  numbor  of  factors  affected  with  the  sign  — , 

1 

(1_Z)-'  =  Y— ^=1+2  +  2"  +  ^^   1   «^+Z^+ 

The  same  result  will  be  obtained  by   applying  the  rule  for  divi- 
sion  (Art.  55). 

1    I    1-2  

1st.  remainder    .     +s  I  l-{-z+2^+z^+z*+  .  .  . 

2d +z' 

3d +z^ 

4th +2" 

+  .  .  . 

2 
Again,  take  the  expression   tt" t5"»  or  2(1—2)-^. 


We  have  ^(1-*)' 


-3 


-3—1  —3—1    —3-2 

2[l-3.(-2)-3.-^-.(-.)^-3.-^— .— ^.(-2)^-.] 

or  2(l-s)-'='=2(l  +  32+622  +  102^  +  15s*+ ) 

To  develop  the  expression  V22  — z^  which  reduces  to 
'•v/2t(l  — ^)'',     we  first  find 


11        5    ^ 

^~V36'^~1848"^ 


hence      V  2.-.^^  V22(l-lz-^2=-g^.^-,  6.c.) 


EXAMPLES. 

1.  To  find  the  value  of   7— -,tj,  or  its  equal  {a+hy''  in  an  m- 
finite  series. 


INDETERMINATE  CO-EFFICIENTS.  241 

2.  To  find  the  value  of ,    in  an  infinite  series. 

r+x 

x^      x^      a:* 
Ans.     r—x-\ -+-tj  ^c. 

3.  Required  the  square  root  of    — ^ — 5-     in  an  infinite  series. 

x^         X*  x^ 

4.  Required  the  cube  root  of    -7-5- — ^-;-     in  an  infinite  series. 


1      7         2*2        5a;*       40a,'«  \ 


Method  of  Indeterminate  Co-efficients.     Recurring  Series 

207.  Algebraists  have  invented  another  method  of  developing 
algebraic  expressions  into  series,  which  is  in  general,  more  simple 
than  those  we  have  just  considered,  and  more  extensive  in  its  appli- 
cations, as  it  can  be  applied  to  algebraic  expressions  of  any  nature 
whatever. 

Before  considering  this  method,  it  will  be  necessary  to  explain 
what  is  meant  by  the  term  function. 

Let  a=:b+c.  In  this  equation,  a,  h  and  c,  mutually  depend  on 
each  other  for  their  values.     For, 

a=i-\-c,  b=a—c,  and  c=a—b. 

The  quantity  a  is  said  to  be  a  function  of  h  and  c,  i  a.  function  of 
a  and  c,  and  c  a  function  of  a  and  b.  And  generally,  wJien  one 
quantity  depends  on  others  for  its  value,  it  is  said  to  be  a  function  of 
those  quantities  on  which  it  depends 

In  order  to  give  some  idea  of  this  method  of  development,  we  will 

a 
suppose  it  is  required  to  develop  the  expression       ,        _   into  a  se- 
ries arranged  according  to  the  ascending  powers  of  x.     It  is  plain 

21 


242  ALGEBRA. 

a 
that  the  expression  can  be  developed ;  for    — — r—    reduces  to 

fl(a'+S'x)~* ;  and  by  applying  the  binomial  formula  to  it,  we  should 
evidently  obtain  a  series  of  terms  arranged  according  to  the  ascend- 
ing powers  of  x.     We  may  therefore  assume 

the  co-efficients  A,  B,  C,  D,  .  .  .  being  functions  of  a,  a',  V ,  but  in- 
dependent of  a;,  it  is  required  to  determine  these  co-efRcients,  which 
are  called  indeterminate  co-efficients. 

For  this  purpose,  multiply  both  members  of  the  equation  (1)  by 
by  a'-\-h'x;  arranging  the  i-esult  with  reference  to  the  powers  of 
X,  and  transposing  a,  it  becomes 

Aa'+Ba'  I  x+Ca'  I  x^'+Da'  I  x^^Ea'  I  x^+  .  .  .  (2). 


^-  i  -a+Ab'  1     -{-Bb'  I      +Cb'   I      +Db' 

Now  if  the  values  of  A,  B,  C,  D,  .  .  .  were  determined,  the 
equation  (1)  would  be  satisfied  by  any  value  given  to  x  ;  this  must 
therefore  be  the  case  also  in  the  equation  (2). 

But  by  supposing  x—0,  this  equation  becomes, 
0  =  Aa'  —  a; 

Whence  A=—r; 

a 

a 
A  being  equal  to  — ;-,  when  x—0,  this  must  be  the  value  of  it  when 

a;  is  any  quantity  whatever,  since  A  is  independent  of  a;  by  hypothe- 
sis ;  therefore  whatever  may  be  the  value  of  x,  the  equation  (2) 
reduces  to 

(       Ba'  I  x-\-Ca'  I  x^-\-Da'  I  a;^+ ;  or,  dividing  by  x, 

^^  i  +Ab'  I     +Bb'  I      +Cb'\ 

_  (       Ba'  \     -\-Ca'  I  X  -{-Da'  I  x^-ir (3). 

^~  I  +Ah'  I     +Bb'  I      +Ch'\ 

This  equation  being  also  satisfied  by  any  value  for  x,  by  making 
a;=:0,  it  becomes  Ba' -\- Ab' =^0 . 


INDETERMINATE  CO-EFFICIENTS.  243 

Ah'  a      V  aV 

Whence        B^ —,  or  £=-— X— =--7^- 

As  this  must  be  the  value  of  B  whatever  may  be  that  of  .r,  we 
will  suppress  the  first  term  Ba'+AV  of  the  equation  (3),  which 
this  value  of  B  makes  equal  to  zero,  and  divide  by  a; ;  it  thus  be- 
comes 

Ca'+Da!  I  x-\-Ea'  I  o^-{-  .  .  . 


-{-Bb'  +  Cb'  I     +Db' 
Making  x=0,  there  results 

Ca'+Bb'  =  0. 
Bb'  I     ab'\      V       ab'' 


X-r=- 


Whence  C— ;— ,  or  C—  —  I y^ 

a  \     a~ 

In  the  same  way  we  should  find 

Da'  +  Ch'=0, 

Cb'          ^      aV^          V            ab'^ 
Whence  Z?= ;— ,  or  Z?= — pr-X  — -r= rr-  ;  and  so  on. 

It  is  easily  perceived  that  any  co-efficient  is  formed  from  that 

b' 
which  precedes  it,  by  multiplying  by ;-;  therefore  we  have, 

a  a      aV        ab'^  ab'^  ab'^ 

—mr-^—r 7oX-\ — 7^^ tt^^-^ — 7^^'*—  •  •  • 

a+bx      a       a-         a^  a*  a^ 

208.  By  reflecting  upon  the  preceding  reasoning,  we  perceive, 
that  the  fundamental  principle  of  the  method  of  indeterminate  co- 
efficients,  depends  upon  this,  viz.,  when  an  equation  of  the  form 
Q=M-\-Nx+Px^-\-Q3?-\-  .  .  .  (M,  N,P,Q,...  being  independent 
of  x),  is  verified  by  any  value  of  x  whatever,  each  of  the  co-efficients 
must  necessarily  be  equal  to  0. 

For  since  these  co-efficients  ai'e  independent  of  x,  when  they  are 
determined  by  any  particular  hypothesis  made  with  respect  to  x, 
the  values  must  answer  for  any  value  of  x  whatever.  Now,  mak- 
ing x=0,  we  find  M=0,  and  dividing  the  equation  by  x,  it  reduces 
to 

0=N+Px+Qx'+  .  .  . 


244  ALGEBRA. 

making  x=0  in  this  equation,  it  becomes  N=0,  and  dividing  the 
equation  by  a:,  it  reduces  to  0=P+Qa;+  .  .  .  and  so  on.  Hence  we 
have 

31=0,  N=0,  P=0,  Q=0  .  .  .  ; 

in  this  manner  we  obtain  as  many  equations  as  there  are  co-effi- 
cients to  be  determined. 

This  principle  may  be  enunciated  in  another  mamier,  viz. 

When  an  equation  of  the  form 

a+hx-\-cx'^-\-do^-\-  .  .  .  z=a' -\-b'x-\-c3?-\-(jl!x^-\-  .  .  . 
is  satisfied  by  any  value  given  to  x,  the  terms  involving  the  same 
powers  in  the  two  members  are  respectively  equal ;  for,  by  trans- 
posmg  all  the  terms  into  the  second  member,  the  equation  will  take 
the  form  0=M+Pa;+Qa;-+  .  .  .  ,  whence 

a— a=:0,  b'  —  h=0,  c'  —  c=0  .  .  .  .  , 

and  consequently, 

dr^a,  h' ^=h,  c'^=L,  d'=^d  .  .  .  ., 
Every  equation  in  which  the  terms  are  arranged  with  reference 
to  a  certain  letter,  and  which  is  satisfied  by  any  value  which  can  be 
given  to  this  letter,  is  called  an  identical  equation,  in  order  to  distin- 
guish it  from  a  common  equation,  that  is, -an  equation  which  can  only 
be  satisfied  by  giving  particular  values  to  this  letter. 

209.  The  method  of  indeterminate  co-efficients  requires  that  we 
should  know  the  form  of  the  development,  with  reference  to  the  ex- 
ponents of  X.  The  development  is  generally  supposed  to  be  ar- 
ranged according  to  the  ascending  powers  of  x,  commencing  with 
the  power  x" ;  sometimes,  however,  this  form  is  not  exact ;  in  this 
case,  the  calculus  detects  the  error  in  the  supposition. 

1 

For  example,  develop  the  expression    -g^^-i"- 

Suppose  that    ^^_^^  =:A+Bx+Cx^+D^+  .  .  .  ., 
whence,  by  clearing  the  fraction,  and  arranging  the  terms, 


RECURRING  SERIES.  245 

.-a\     -b\     -c\ 

whence  (Art.  208), 

—  1  =  0,  3^1=0,  2B-A=0 

Now  the  first  equation,  —1=0,  is  absurd,  and  indicates  that  the 

above  form  is  not  a  suitable  one  for  the  expression      g^^_^^  ;     but 

11, 

if  we  put  this  expression  under  the  form    —  X  g3^»    ^"^^  suppose 

that 

—  X^=—{A+Bx  +  ar^+DxP+ ), 

X      3—x      x^ 

it  will  become,  after  the  reductions  are  made, 

3^  +  35  I  x+SC  I  af  +  SD  I  aP-{-  .  .  ., 


0-  I  _i_A    \      -B\        -C 

which  gives  the  equations 

3A  —  1=0,  2B-A=0,  ^C-B=0  .  .  ., 

1111 
whence  A=-,  B^-,  C=-,  D=-  ... 

Therefore, 

-3:^^=t(t+t^'+27'^+8I'^  +  •  •  •)' 

=\--'+^-''+^-+^-'+  •  •  • ' 

that  is,  the  development  contains  a  term  affected  with  a  negative 
exponent. 

Reciwring  Series. 
210.  The  development  of  algebraic  fractions  by  the  method  of 
indeterminate  co-efficients,  gives  rise  to  certain  series,  called  recur- 
ring series. 

A  recurring  series  is  the  development  of  a  rational  fraction  invoU. 
ing  X,  made  according  to  a  fixed  law,  and  containing  the  ascending 
potcers  of  x  in  its  different  terms. 
21* 


246 


It  has  been  shown  in  Art.  207,  that  the  development  of  the  ex. 


pression 


is  the  series  — r- 


aV        ah' 


-3?+  .  . .,  in 


a'  +  b'x  a      a'         a- 

which  each  term  is  formed  by  multiplying  that  which  precedes  it 

by -X. 

a 

This  property  is  not  peculiar  to  the  proposed  fraction  ;  it  belongs 
to  all  rational  algebraic  fractions,  and  it  consists  in  this,  viz. :  Every 
rational  fraction  involving  x,  may  he  developed  into  a  series  of  tenns, 
each  of  which  is  equal  to  the  algehraic  sum  of  the  products  which 
arise  from  multiplying  certain  terms  of  a  particular  expression,  hy 
certain  of  the  preceding  terms  of  the  series. 

The  particular  expression,  from  which  any  term  of  the  series 

may  be  found,  when  the  preceding  terms  are  known,  is  called  the 

scale  of  the  series  ;  and  that  from  which  the  co-efRcient  may  be 

formed,  the  scale  of  the  co-efficients. 

.         h' 
In  the  preceding  series,  the  scale  is yx,  and  the  series  is  call- 

h' 


ed  a  recurring  series  of  the  first  order,  and  — - 

co-efficients. 

a-\-ha 


Let  it  be  required  to  develop 
a+h. 


is  the  scale  of  the 


mto  a  series. 


Assume 


a'  -\-h'x-\-c'OiF 
=A+Bx-\-Cx^+Dx''+Ex''+  .  .  . 


a'  -\-h'x-\-c'xF 
Clearing  the  fraction  and  transposing,  we  have 


0= 


Aa'+Ba' 

x  +  Ca' 

x'+Da' 

x^+Ea' 

-a+Ah' 

+Bh' 

-\-Ch' 

■^Bh' 

-b 

-{-Ac' 

-{-Be' 

^Cc' 

a;^  + 


which  gives  the  equations 


Aa!- 


■  a=0,  or  A=^-^ 
a 


Ba'-{-Ah'-  h 


:0,  or  B= — 7A+-r 
a'        a! 


RECURRING  SERIES.  247 


h'  d 

Ca'+5^'+^c'=rO,  or  C= r^ r^ 

'  a  a 

h'  d 

Da'  +  Ch'-^Bd=0,  or  D= j^—rB 

a  a 

V          d 
Ea'+Db'  +  Cd=0,  or  E= jD jC. 


Whence  we  perceive  that  the  two  first  co-efficients  are  not  ob- 

tained  by  any  law  ;  but  commencing  at  the  third,  each  co-efficient 

is  formed  by  multiplying  the  two  which  precede  it  respectively  by 

h'  d 

r   and  ——7,  viz.  that  which  immediately  precedes  the  requir- 

a  a 

y  d 

ed  co-efficient  by j,  that  which  precedes  it  two  terms  by 7, 

and  taking  the  algebraic  sum  of  the  products.     Hence, 

(-^-4) 

is  the  scale  of  the  co-efficients. 

From  this  law  of  the  formation  of  the  co-efficients,  it  follows  that 
the  third  term  of  the  series,  Cx^  is  equal  to 

V  d 
a            a 

V  d 

or  rxB.x i-xr.A. 

a  a 

In  like  manner,  we  have  for  D^P 

V  d 
jC:,? jB^ 

a!  a 

V  d 

or  7^  •  Cx^ — —rX^  .  Bx. 

a  a 

Hence,  each  term  of  the  required  series,  commencing  at  the 
third,  is  obtained  by  multiplying  the  two  terms  which  precede,  re- 
spectively by 


248  ALGEBRA. 

V        d 

-X -x^, 

a         a! 

and  taking  the  sum  of  the  products :  hence,  this  last  expression  is 
the  scale  of  the  series. 

211.  Recurring  series  are  divided  into  orders,  and  the  order  is 
estimated  by  the  number  of  terms  contained  in  the  scale. 

.  a  . 

Thus,  the  expression  gives  a  recurring  series  of^  the frst 

V 

order,  the  scale  of  which  is -x. 

a 

.  a-\-bx 

The  expression  ■  ,■   y — ] — 7~2  will  give  a  recurring  series  of  the 
a  -\-o  iC-pC  X 

second  order,  of  which  the  scale  will  be 
/      h'  c'     \ 

\     a'  a!     I 

The  series  obtained  in  the  preceding  Art.  is  of  the  second  order. 
In  general,  an  expression  of  the  form 

a-\-l)x-\-cdc^-\-  .  .  .  kx^-'*- 
a'+^»'x+cV+  .  .  .   &V 
gives  a  recurnng  series  of  the  n""  order,  the  scale  of  which  is 
V 


(0            c                    k     \ 
yX, tSc"  .  .   . -X"). 
a            a                   a     / 


Remakk.  It  is  here  supposed  that  the  degree  of  x  in  the  numera- 

tor  is  less  than  it  is  in  the  denominator.     If  it  was  not,  it  would  first 

be  necessary  to  perform  the  division,  arranging  the  quantities  with 

reference  to  x,  which  would  give  an  entire  quotient,  plus  a  fraction 

similar  to  the  above. 

_        .     ,  .       l—x—Sa^-\-4:X^+x* 

Thus,  in  the  expression      ^_^^^_^^^,_^,   • 

a;4+4ar'-3ic2-a?+l  )    —x^+S.'c'—5x+2 


+lx^—8x^+x        )    —x—7 


+  13.t--34a,  +  15. 


PROGRESSIONS  BY  DIFFERENCES.  249 

Performing  the  division,  we  find  the  quotient  to  be  —x—1,  plus 
the  fraction. 

13.i^__34a:+15  Ib-Mx  +  lZt? 

-,  or 


_a;3  +  3x^-5x  +  2  '  2—bx+Zx^ 


CHAPTER  V. 


Of  Progressions^  Continued  Fractions^  and 
Logarithms. 

212.  This  chapter  is  naturally  connected  with  the  last,  as  it  ex- 
plains the  properties  of  two  kinds  of  series,  and  also  presents  an  ap- 
plication of  the  theory  of  exponents.  It  moreover  completes  that 
part  of  algebra  which  is  absolutely  necessary  for  the  study  of 
Trigonometry,  and  the  Application  of  Algebra  to  Geometry. 

Progressions  by  Differences. 

213.  A  progression  ly  differences,  or  an  Arithmetical  progression, 
is  a  series  m  which  the  successive  terms  continually  increase  or  de- 
crease by  a  constant  quantity,  which  is  called  the  common  difference 
of  the  progression. 

Thus,  in  the  two  series 

1,     4,     7,  10,  13,   16,  19,  22,  25.  .  .  . 
60,  56,  52,  48,  44,  40,  36,  32,  28.  .  .  . 
The  first  is  called  an  increasing  progression,  of  which'  the  com- 
mon difference  is  3,  and  the  second  a  decreasing  progression,  of 
which  the  common  difference  is  4. 

214.  If  there  are  four  quantities  a,  h,  c,  d,  in  arithmetical  pro- 
gression, a  is  said  to  be  to  h,  as  c  to  d  :  and  a  and  c  are  called  ante, 
cedents,  and  b  and  d  consequents. 


250  ALGEBRA. 

In  general,  let  a,  b,  c,  d,  e,f,  .  .  .  designate  the  terms  of  a  pro- 
gression by  differences  ;  it  has  been  agreed  to  write  them  thus  : 
a.b  .c  .  d.  e  .f.  g  .  h  .  i .  k.  .  ,  , 

This  series  is  read,  a  is  to  b,  as  b  is  to  c,  as  c  is  to  d,  as  d  is  to  e, 
ifec.  or  a  is  to  b,  is  to  c,  is  to  d,  is  to  e,  &c.  This  is  a  series  of  con- 
tinued equi-differences,  in  which  each  term  is  at  the  same  time  a  con- 
sequent  and  antecedent,  with  the  exception  of  the  first  term,  which 
is  only  an  antecedent,  and  the  last,  which  is  only  a  consequent. 

215.  Let  r  represent  the  common  difference  of  the  progression, 
which  we  will  consider  as  increasing.  In  the  case  of  a  decreasing 
progression,  it  will  only  be  necessary  to  change  r  into  —r,  in  the  re- 
suits. 

From  the  definition  of  the  progression,  it  evidently  follows  that 
b—a-\-r,  c=b-\-r=^a+2r,  d=c-\-r=a+3r  : 

and  in  general,  any  term  of  the  series  is  equal  to,  the  first  plus  as 
many  times  the  common  difference  as  there  are  preceding  terms. 

Thus,  let  I  be  any  term,  and  n  the  number  which  marks  the  place 
of  it,  the  expression  for  this  general  term,  is 
l^a  +  {n—l)r. 

That  is,  the  last  term  is  equal  to  the  first  term,  plus  ike  product  of 
the  common  difference  by  the  number  of  terms  less  one. 

If  we  suppose  n  successively  equal  to  1,  2,  3,  4,  &c.  we  shall  ob- 
tain the  first,  second,  third,  fourth,  &c.  term  of  the  progression. 

The  formula  Z=a+(n—l)r,  serves  to  find  any  term  whatever, 
without  our  being  obliged  to  determine  all  those  which  precede  it. 

Thus,  by  making  n=50,  we  find  the  50"*  term  of  the  progres- 
sion, 

1.4.7.10.13.16.19....  Z=l+49x3  =  148.   ~ 

216.  If  the  progression  were  a  decreasing  one,  we  should  have 

Z=a— (n— l)r. 
That  is,  in  a  decreasing  arithmetical  progression,  the  last  term  is 


PROGRESSIONS  BY  DIFFERENCES.  251 

equal  to  the  first  term  minus  the  product  of  the  common  difference  hy 
the  number  of  terms  less  one. 

217.  A  progression  by  differences  being  given,  it  is  proposed  to 
prove  that,  the  sum  of  any  two  terms,  taken  at  equal  distances  from 
the  two  extremes,  is  equal  to  the  sum  of  the  two  extremes. 

Let  a.h  .c  .d.e.f .  .  .  .  i.k.l,  be  the  proposed  progression, 
and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denotes  a  term  which  has  p  terms 
before  it,  and  y  a  term  which  has  p  terms  after  it,  we  have,  from 

what  has  been  said, ir=a+px^> 

and y=l—pxr; 

whence,  by  addition, x-\-y=a-{-l. 

which  demonstrates  the  proposition. 

This  being  the  case,  write  the  progression  below  itself,  but  in  an 
inverse  order,  viz. 

a.h  .c.d.e  .f  .   .   .  .  i.k.l. 
I  .k  .i c.b  .a. 

Calling  S  the  sum  of  the  terms  of  the  first  progression,  2S  will 
be  the  sum  of  the  terms  in  both  progressions,  and  we  shall  have 
2S={a  +  l)  +  {b+k)  +  {c+i)  .  .  .  +{i+c)  +  (k  +  b)  +  (l+a); 

or,  since  the  number  of  the  parts  a-\~l,  b-\-k,  c-{-i is  equal 

to  n, 

2S=ia+l)n,  or  sJ-^^-^^ 

That  is,  the  sum  of  a  progression  by  differences,  is  equal  to  half 
the  sum  of  the  two  extremes,  multiplied  by  the  number  of  terms. 

If  in  this  formula  we  substitute  for  I  its  value,  a  +  (n— l)r,  we 
obtain 

[2a  +  (n-l)r]n    . 
S= , 

but  the  first  expression  is  the  most  useful. 

{a+l)n 
218.   The   formulas   Z=a4-(n— l)r,    S= — - — ,    contain   five 


252  ALGEBRA. 

quantities,  a,  r,  n,  I  and  S,  and  consequently  give  rise  to  the  follow- 
ing general  problem,  viz. :  Any  three  of  these  jive  quantities  being 
given,  to  determine  the  other  two. 

There  will,  therefore,  be  as  many  different  cases  as  there  can  be 
formed  combinations  of  five  letters  taken  three  in  a  set :  that  is, 

5_1    5-2 
5.— —.-^-=10.  (Art.  163). 
2  o 

Of  these  cases  we  shall  consider  only  the  most  important. 
We  already  know  the  value  of  S  in  terms  of  a,  n  and  r. 
From  the  formula  l=a+{n—l)r,  we  find 

a=l—{n—l)r. 
That  is,  the  first  term  of  an  increasing  arithmetical  progression  is 
equal  to  the  last  term,  minus  the  product  of  the  common  difference  by 
the  number  of  terms  less  one. 

From  the  same  formula,  we  also  find, 
l-a 

That  is,  in  any  arithmetical  progression,  the  common  difference  is 
equal  to  the  difference  between  the  two  extremes  divided  by  the  number 
of  terms  less  one. 

219.  The  last  principle  affords  a  solution  to  the  following  ques- 
tion. 

To  find  a  number  m  of  arithmetical  means  bettveen  tioo  given  mun- 
hers  a  and  b. 

To  resolve  this  question,  it  is  first  necessary  to  find  the  common 
difference.  Now  we  may  regard  a  as  the  first  term  of  an  arith. 
metical  progression,  b  as  the  last  tei-m,  and  the  required  means  as 
intermediate  terms.  The  number  of  terms  of  this  progression  will 
be  expressed  by  ??i+2. 

Now,  by  substituting  in  the  above  formula,  b  for  /,  and  wi+2 

h—a  b—a  . 

for  n,  it  becomes  r= — r>  or   ^= TT"  5  that  is,  the   com. 

wt+2  — 1  wi+1 

mon  difference  of  the  required  progression  is  obtained  by  dividing 


PROGRESSIONS  BY  DIFFERENCES.  253 

the  difference  between  the  given  numbers  a  and  b,  by  one  more 
than  the  required  number  of  means. 

Having  obtained  the  common  difference,  form  the  second  term  of 
the  progression,  or  the  Jirst  arithmetical  mean,  by   adding  r,  or 

:r->  to  the  first  term  a.     The  second  mean  is  obtained  by  aug. 

m  +  1 

menting  the  first  by  r,  &c. 

For  example,  let  it  be  required  to  find  12  arithmetical  means  be- 

77-12     65 
tween  12  and  77.     We  have  r= — — — =—=5,  which  gives  the 

lo  xO 

progression     12  .  17  .  22  .  27  ...  72  .  77. 

220.  Remark.  If  the  same  number  of  arithmetical  means  are 
inserted" between  all  of  the  terms,  taken  two  and  two,  these  terms, 
and  the  arithmetical  means  united,  will  form  but  one  and  the  same 
progression. 

For,  let  a  .  b  .  c .  d .  e ./ .  .  .  .  he  the  proposed  progression,  and 
m  the  number  of  means  to  be  inserted  between  a  and  b,  b  and  c, 
c  and  d  .  .  . 

From  what  has -just  -been  said,  the  common  difference  of  each 

b—a     c—b    d—c 

partial  progression  will  be  expressed  by  — ,  -, .  .  .  ., 

*^  ^    °  r  ^   m  +  1'  m+1    7n  +  l 

which  are  equal  to  each  other,  since  a,  b,  c  .  .  .  are  in  progression : 
therefore,  the  common  difference  is  the  same  in  each  of  the  partial 
progressions ;  and  since  the  last  term  of  the  first,  forms  the^r*^  term  of 
the  second,  <Sic.  we  may  conclude  that  all  of  these  partial  progres- 
sions form  a  single  progression. 

EXAMPLES. 

1.  Find   the   sum  of  the  first   fifty  terms   of  the  progression 
2.9. 16 . 23  .  .  . 
For  the  50th  term  we  have  Z=2  +  49x 7  =  345. 


Hence  6^=(2-f-345)Xy  =  347x25=8675. 


50 
22 


254  ALGEBRA. 

2.  Find  the  100th  term  of  the  series     2  .  9  .  16  .  23  .  .  . 

Arts.     695. 

3.  Find  the  sum  of  100  terms  of  the  series     1.3.5.7.9... 

Ans.     10000. 

4.  The  greatest  term  is  70,  the  common  difference  3,  a'nd  the 
number  of  terms  21,  what  is  the  least  term  and  the  sum  of  the 
series  ?  Ans.     Least  term  10  :  sum  of  series  840. 

5.  The  first  term  of  a  decreasing  arithmetical  progression  is  10, 

the  common  difference  — -,  and  the  number  of  terms  21  :  required 
the  sum  of  the  series.  Ans.     140. 

6.  In  a  progression  by  differences,  having  given  the  common  dif- 
ference  6,  the  last  term  185,  and  the  sum  of  the  terms  2945,  find 
the  first  term,  and  the  number  of  terms. 

A71S.     First  term  =5,  number  of  terms  31. 

7.  Find  9  arithmetical  means  between  each  antecedent  and  con- 
sequent  of  the  progression      2.5.8.11.14.  .  . 

Ans.     Ratio,  or  r=  0,3. 

8.  Find  the  number  of  men  contained  in  a  triangular  battalion, 

the  first  rank  contaming  1  man,  the  second  2,  the  third  3,  and  so 

on  to  the  n"*,  which  contains  n.     In  other  words,  find  the  expression 

for  the  sum  of  the  natural  numbers  1,  2,  3  .  .  .,  from  1  to  n,  inclu- 

n(n+l) 
sively.  Ans.     S= — . 

9.  Find  the  sum  of  the  n  first  terms  of  the  progression  of  uneven 
numbers  1,  3,  5,  7,  9  .  .  .  Ans.     S=n'. 

10.  One  hundred  stones  being  placed  on  the  ground,  in  a  straight 
line,  at  the  distance  of  2  yards  from  each  other :  how  far  will  a  per- 
son travel,  who  shall  bring  them  one  by  one  to  a  basket,  placed  at 
2  yards  from  the  first  stone  ? 

Ans.     11  miles,  840  yards. 


PROGRESSIONS  BY  QUOTIENTS.  255 

Geometrical  Progression,  or  Progressions  hy  Quotients. 

221.  A  Geometrical  progression,  or  progression  by  quotients,  is  a 
series  of  terms,  each  of  which  is  equal  to  the  product  of  that  which 
precedes  it,  by  a  constant  number,  which  number  is  called  the  ratio 
of  the  progression  ;  thus  in  the  two  series  : 

3,     6,  12,  24,  48,  96  .  .  . 

64,  16,     4,     1,  |,  ]^  .  .  . 

each  term  of  the  first  contains  that  which  precedes  it  twice,  or  is 
equal  to  double  that  which  precedes  it ;  and  each  term  of  the  second 
is  contained  in  that  which  precedes  it'  four  times,  or  is  a  fourth  of 
that  which  precedes  it ;  they  are  then  progressions  by  quotients,  of 

which  the  ratio  is  2  for  the  first,  and    —    for  tlie  second. 

Let  a,  b,  c,  d,  e,f .  .  .  be  numbers  in  a  progression  by  quotients, 
they  are  written  thus  ;  a:b:c:d:eif:g...,  and  it  is  enun- 
ciated  in  the  same  manner  as  a  progression  by  differences  ;  however 
it  is  necessary  to  make  the  distinction  that  one  is  a  series  of  equal 
differences,  and  the  other  a  series  of  equal  quotients  or  ratios,  in  which 
each  term  is  at  the  same  time  an  antecedent  and  a  consequent,  ex- 
cept the  first,  which  is  only  an  antecedent,  and  the  last,  which  is  only 
a  consequent. 

222.  Let  q  denote  the  ratio  of  the  progression  a  :  b  :  c  :  d  .  .  ., 
q  being  >1  when  the  progression  is  increasing,  and  $'<1  when  it  is 
decreasing  :  we  deduce  from  the  definition,  the  following  equalities. 

b=aq,     c=bq=aq^,     d=cq—aq^,     e=dq=aq* .  ,  . 
and  in  general,  any  term  7i,  that  is,  one  which  has  n  — 1  terms  be- 
fore it,  is  expressed  by  aq"''^. 

Let  I  be  this  term ;  we  have  the  formula  l=aq''~^,  by  means  of 
which  we  can  obtain  the  value  of  any  term  without  being  obliged 
to  find  the  values  of  all  those  which  precede  it.  That  is,  the  last 
term  of  a  geometrical  progression  is  equal  to  the  first  term  mvitiplied 


by  the  ratio  raised  to  a  "power  whose  exponent  is  one  less  than  the 
number  of  terms. 

For  example,  the  8*  term  of  the  progression  2":  6^  18  :  5C  . ., 
is  equal  to   2x3^=2x2187=4374. 

In  like  manner,  the  12*  term  of  the  progression     .     ^    .     .     ., 

64  :  16  :  4  :  1  :  —  ...  is  equal  to 


64(-i)"=!l=l=_^. 
V4/        4"     4»       65536 


223.  We  will  now  proceed  to  determine  the  sum  of  n  terms  of 
the  progression  a  :  b  :  c  :  d  :  e  :f:  .  .  ..:i:k:ly  I  denoting  the 
n""  term. 

We  have  the  equations  (Art.  222), 

b=aq,     c=hq,     d=cq,     e=dq,  .  .  .  k=iq,     l=kq; 
and  by  adding  them  all  together,  member  to  member,  we  deduce 

b+c+d+e+  .  .  .  +k  +  I={a  +  h  +  c+d+  .  .  .  +i-{-k)q; 
or,  representing  the  required  sum  by  S, 

S.-ra  =  {8-l)q=Sq-]q,     or     Sq-S=lq-a', 

Iq  —  a 

whence  S  = —] 

q-l 

That  is,  to  obtain  the  sum  of  a  certain  number  of  terms  of  a  pro- 
gression by  quotients,  inultijyly  the  last  term  by  the  ratio,  subtract  the 
first  term  from  this  product,  and  divide  the  remainder  by  the  ratio  di. 
vtinisJied  by  unity. 

When  the  progression  is  decreasing,  we  have  5<1   and  Z<a ; 

a  —  lq 
the  above  formula  is  then   written  under  the   form     S  =  ^— — , 

in  order  that  the  two  terms  of  the  fraction  may  be  positive. 

By  substituting  o?"'*  for  I  in  the  two  expressions  for  S,  they  be- 

agi" — a  ^     a  —  aq" 

oome,  S= r— ,  and  6=-^; . 

5-1  l~q 


FROGRESSIOIVS  BY  QUOTIENTS.  957 

EXAMPLES. 

1.  Find  the  first  eight  terms  of  the  progression 

2  :  6  :  18  :  54  :  162  .  .  .  :  2x3"'  or  4374 

^       Iq-a       13122-2 

S=^— — = =6560. 

q—1  2 

2.  Find  the  sum  of  the  first  twelve  terms  of  the  progression 

1  /1\"  1 

64  :  16  :  4  :  1  :  —  :  .  .  .  :  64| 


(I)-'- 


4 V  4  /    '  65536 ' 


1  1  1 

64 — 7r:rr—X^      256- 


„       a—lq                65536       4                    65536  65535 

S=   ,     „   = s = ^ =85+ 


3  3  196608 

T 

We  perceive  that  the  principal  difficulty  consists  in  obtaining  the 
numerical  value  of  the  last  term,  a  tedious  operation,  even  when 
the  number  of  terms  is  not  very  great. 

3.  What  debt  may  be  discharged  in  a  year,  or  twelve  months, 
by  paying  $1  the  first  month,  ^2  the  second  month,  $4  the  third 
month,  and  so  on,  each  succeeding  payment  being  double  the  last ; 
and  what  will  be  the  last  payment  ? 

Ans.     Debt,  $4095  :  last  payment,  #2048. 

224.  Remark.    If,  in  the  formula   S= — ,    we  suppose 

0 

5-— 1,  it  becomes  S=—. 

This  result,  which  is  sometimes  a  symbol  of  indetermination,  is 
also  often  a  consequence  of  the  existence  of  a  common  factor 
(Art.  113),  which  becomes  nothing  by  making  a  particular  hypo- 
thesis  respecting  the  given  question.  This,  in  fact,  is  the  case  in 
the  present  question  ;  for  the  expression  ^"—1  is  divisible  by  q—1, 
(Art.  59),  and  gives  the  quotient 

5-1 +^--"+^"-3+  ...  +5+1;. 
22* 


2p8  ALGEBRA. 

hence  the  value  of  S  takes  the  form 

<S=a^"-'+a?"-^+C5"-^+  .  .  .  +aq-\-a. 

Now,  making  5=1,  we  have  S=a+a-[-a+  .  .  .  +a=na. 

We  can  obtain  the  same  result  by  going  back  to  the  proposed 
progression,  a  :  b  :  c,:  .  •-•  -l,  which,-in  the  particular  case  of  j=l, 
reduces  to  a  :  a  :  a  :  .  .  .  :  a,  the  sum  of  which  series  is  equal 
to  na. 

0       . 

The  result  — ,  given  by  the  formula,  may  be  regarded  as  indi- 
cating that  the  series  is  characterized  by  some  particular  property. 
In  fact,  the  progression,  being  entirely  composed  of  equal  terms,  is 
no  more  a  progression  by  quotients  than  it  is  a  progression  by  diffe- 
rences.    Therefore,  in  seeking  for  the  sum  of  a  certain  number  of 

aiq"  —  !) 
the  terms,  there  is  no  reason  for  using  the  formula    S= •- — , 

{a.+l)n,. 
in  preference  to  the  formula  S= — - — ,  which  gives  the  sum  in 

the  progression  by  differences. 

Of  Progressions  having  an  infinite  number  of  terms. 

225.  Let  there  be  the  decreasing  progression  a:b:c:d   e  :f: . . ., 

a — aq" 
containing  an.  indefinite  number  o£  terms.  The  formula  S=— , 

which  represents  the  sum  of  n  of  its  terms,  can  be  put  under  the 

a         aq" 

form  *S=- . 

1-q     1-q 

Now,  since  the  progression  is  decreasing,  5'  is  a  fraction ;  and  q'' 
is  also  a  fraction,  which  diminishes  as  71  increases.     Therefore  the 

greater  the  number  of  terms  we  take,  the  more  will Xq" 

diminish,  and  consequently,  the  more  will  the  partial  sum  of  these 
terms  approximate  to  an  equality  with  the  first  part  of  S,  that  is,  to 


PROGRESSIONS  BY  QUOTIENTS.  25>9 


.     Finally,  when  n  is  taken  greater  than  any  given  number, 

a 
or  n  =  00,  then    - — ^X<r     will  be  less  than  any  given  number, 

a 

or  will  become  equal  to  0  ;  and  the  expression will  repre- 
sent the  true  value  of  the  sum  of  all  the  terms  of  the  series. 

Whence  we  may  conclude,  that  the  expression  for  the  sum  of  the 
terms  of  a  decreasing  progression,  in  which  dhe  number  of  terms  is 
infinite,  is 

This  is,  properly  speaking,  the  limit  to  which  ihe  ■partial  sum^  ap- 
proach, by  taking  a  greater  number  of  terms  in  the  progression. 

a 

The  difference  between  these  sums  and can  become  as 

\-q 

small  6is  we  please,  and  will  only  become  nothing  when  the  number 

of  terms  taken  is  infinite. 


EXAMPLES. 

1.  Find  the  sum  of 

1111 

We  have  for  the  expression  of  the  sum  of  the  terms 
a  1  3 

3 

The  error  committed  by  taking  this  expression  for  the  value  of 
the  sum  of  the  n  first  terms,  is  expressed  by 

1-j'^       2\3/' 
First  take  n=5  ;  it  becomes 


\  3  /       2.3*      162 


260  ALGEBRA. 

When  n=6,  we  find 

3  / 1  v<'_    1        1  _    1 
TxY/  ~  162  •  ~3~  486  • 


3 

Whence  we  see  that  the  error  committed,  when  —  is  taken  for 

the  sum  of  a  certain  number  of  terms,  is  less  in  proportion  as  this 
number  is  greater. 

A-gain  take  the  progression 

11111 
^  '"2''T'¥=16-32-  '^'^ 

We  have  S=- = =2. 

226.  The  expression  8=- ,  can  be  obtained  directly   from 

the  progression   a  :  h  :  c  :  d  :  e  if :  g  :  .  .  ^ 

For,  take   the   equations  b=aq,  c=bq,  d=cq,  e=dq of 

which  the  number  is  indefinite,  and  add   them  together,  member  to 
member ;  we  have 

h-{-c+d-{-e+  .  .  .  ={a+b  +  c+d-^  •••)?• 

Now,  the  first  member  is  evidently  the  proposed  series,  diminish- 
ed by  the  first  term  a  ;  it  is  therefore  expressed  by  S  — a  ;  the  se- 
cond member  is  q  multiplied  by  the  entire  series,  since  there  is  no 
last  term,  or  rather  this  last  term  is  nothing ;  hence  the  expression 
for  this  member  is  qS,  and  the  above  equality  becomes  S  —  a=qS, 

a 
whence  S=- . 

1-q 

a 
In  fact,  by  developing into  a  series  by  the  rule  for  divi- 
sion, we  shall  find  the  result  to  be  rt-[-^?+f'?^+^?^+  •  •  •>  which 
is  nothing  more  than  the  proposed  series,  having  b,  c,  d  .  .  .  replaced 
by  their  values  in  functions  of  a. 


PROGRESSIONS  B-Y  QUOTIENTS.  f^V 

a 
227.  When  the  series  is  increasing,  the  expression  cannot 

be  considered  as  Or  limit  of  the  partial  sums ;  because,  the  sum  of  a 

a         aq" 
given  number  of  terms  being  S=t— —  —  j-^,    (Art.   225),   the 

second  part augments  numerically,  in  proportion  to  the  in- 
crease of  n ;  hence  the  greater  the  number  of  terms  teken,  the 

a 
more  the  expression  of  their  sum  will  differ  numerically  from  ■— — . 

a 
The  formula  S=:j is,  in  this  case,  merely  the  algebraic  ex- 
pression which,  by  its  development,  gives  the  series- 
a+aq+aq'^+aq^  .  .  . 
There  is  another  circumstance  presents  itself  here,  which  appears 

very  singular  at  first  sight.     Since is   the   fraction    which 

generates  the  above  series,  we  should  have 

j—=a  +  aq+aq''-{-aq^+aq^-{-  .  .  . 

Now,  by  making  a=l,  q  —  2,  this  equality  becomes 

y— g.  or  -l^l+2.+  4+8  +  16+32+  .  .  . 

an  equation  of  which  the  first  member  is  negative  whilst  the  second 
is  positive,  and  greater  in  proportion  to  the  value  of  q. 

To  interpret  this  result,  we  will  observe  that,  when  in  the  equa- 
tion    =:a+aq-{-aq''-\-aq^-{-  .  .  .,  we  stop  at  a  certain  term  of 

the  series,  it  is  necessary  to  complete  the  quotient  in  order  that  the 
equality  may  subsist.  Thus,  in  stopping,  for  example,  at  the  fourth 
term,  aq^. 


ALGEBRA. 

a 

+  aq 

+    0^^ 

1-q 

Ist  remainder 
2d. 

aq* 

a+aq+aq^+aq^+j— 

3d. 

+  aq' 

4th. 

+   aq' 

aq^ 
It  is  necessary  to  add  the  fractional  expression to  the  quo- 

lient,  which  gives  rigorously, 

a  aq* 

——=a  +  aq+aq^+aq=+YZ:^' 

If  in  this  exact  equation  we  make  a=l,  q=2,  it  becomes 

16 

-1  =  1+2  +  4+8+— Y=l+2  +  4  +  8-16, 

which  verifies  itself. 

In  general,  when  an  expression  involving  x,  designated  by 
/(x),  which  is  called  a  function  of  x,  is  developed  into  a  series 

of  the  foiTn  a-]-ix-\-ca^-\-dx^-{- ,   we  have  not  rigorously 

f(x)=a-\-bx-{-cx^-{-dx^-}-  .  .  .,  unless  we  conceive  that,  in  stopping 
at  a  certain  term  in  the  second  member,  the  series  is  completed  by  a 
certain  expression  involving  x. 

When,  in  particular  applications,  the  series  is  decrea,s«?^  (Art.  203  j, 
the  expression  which  serves  to  complete  it  may  be  obtamed  as  near 
as  we  please,  by  prolongmg  the  series  ;  but  the  contrary  is  the  case 
when  the  series  is  increasing,  for  then  it  must  not  be  neglected. 
This  is  thereason  why  increasmg  series  cannot  be  used  for  approxi- 
mating to  the  value  of  numbers.  It  is  for  this  reason,  also,  that 
algebraists  have  called  those  series  which  go  on  diminishing  from 
term  to  term,  converging  series,  and  those  in  which  the  terms  go  on 
increasing,  diverging  series.  In  the  first,  the  greater  the  number 
of  terms  taken,  the  nearer  the  sum  approximates  numerically  to  the 
expression  of  which  this  series  is  the  development ;  whilst  in  the 
others,  the  more  terms  we  take,  the  more  their  sum  differs  from  the 
numerical  value  of  this  expression. 


FROORESSIONS  BY  QUOTIENTS.  263 

228.  The  consideration  of  the  five  quantities  a,  q,  n,  I  and  S, 

Iq — a 
which  enter  in  the  two  formulas  l=aq''-^,  S= —  (Arts.  222  & 

223),  gives  rise  to  ten  problems,  as  in  the  progression  by  differences 
(Art.  218).  Of  these  cases,  we  shall  consider  here,  as  we  did 
there,  only  the  most  important.  We  will  first  find  the  values  of  S 
and  q  in  terms  of  a,  I  and  n. 

I  "-'  /  Z 

The  first  formula  gives  q"^^= — ,  whence  q=  \/ — .  Substi- 
tuting this  value  in  the  second  formula,  the  value  of  S  will  be  ob- 
tained. 

"^^   /  I 
The  expression  q=  \/  —  furnishes  the  means  for  resolving  the 

following  question,  viz. 

To  find  m  mean  proportionals  between  two  given  numbers  a  and 
b ;  that  is,  to  find  a  number  m  of  means,  which  will  form  with  a 
and  b,  considered  as  extremes,  a  progression  by  quotients. 

For  this  purpose,  it  is  only  necessary  to  know  the  ratio ;  now  the 

required  number  of  means  being  m,  the  total  number  of  terms  is 

equal  to  m  +  2.     Moreover,  we  have  Z=&,  therefore  the  value  of  q 

"+'     /Z» 
becomes  9=     \/  —  ;  that   is,  we  must  divide  one  of  the  given 

numbers  (b)  by  the  other  (a),  then  extract  that  root  of  the  quotient 

wJwse  index  is  one  more  than  the  required  number  of  means. 

Hence,  the  progression  is 

"'+1      /  ])        "•+!      /Jr        ">+!      /P 
a  -.a     \/---a     V  ^  :  «     s/ ^  :  .  .  .  b. 

Thus,  to  insert  six  mean  proportionals  between  the  numbers  3 

'    /  384      7  / 

and  384,  we  make  ?tt=6,  whence  q=K/  — — — =  y  128  =  2  ; 

whence  we  deduce  the  progression 

3  :  6  :  12  :  24  :  48  :  90  :  192  :  384. 
Remark.   When  the  same  number  of  mean  proportionals  are  in- 


264  ALOEBR^. 

serted  between  all  the  terms  of  a  progression  by  quotients,  taken  two 
and  two,  all  the  progressions  thus  formed  will  constitute  a  single  pro- 
gression. 

229.  Of  the  ten  principal  problems  that  may  be  proposed  in 
progressions, /our  are  susceptible  of  being  easily  resolved.  The 
following  are  the  enunciations,  with  the  formulas  relating  to  them. 

1st.  a,  q,  n,  being  given,  to  find  I  and  S. 

l^a  "-»•  S-  ^^""^  ^"^^"^-^^ 
2d.  a,  n,  I,  being  given,  to  find  q  and  S. 


3d.  q,  n,  I,  being  given,  to  find  a  and  -S*. 

4th.  q,  n,  S,  being  given  to  find  a  and  I. 

S{q-1)  Sq'^-\q-l) 

,    I- 


r-i  '         9"-i 

Two  other  problems  depend  upon  the  resolution  of  equations  of 
a  degree  superior  to  the  second  ;  they  are  those  in  which   the  un- 
known  quantities  are  supposed  to  be  a  and  q,  or  I  and  q. 
For,  from  the  second  formula  we  deduce 

a—lq  —  Sq+S; 
Whence,  by  substituting  this  value  of  a  in  the  first  l=aq"'^, 
l^{lq-Sq  +  S)q'-\ 
or,  (S-l)q-'-Sq''~'+l=^0. 

an  equation  of  the  7i"'  degree. 

In  like  manner,  in  determining  I  and  q,we  should  obtain  the  equa- 
tion  aq"—Sq  +  S  —  a=0. 

230.  Finally,  the  other  four  problems  lead  to  the  resolution  of 


CONTINUED  FRACTIONS  265 

equations  of  a  peculiar  nature ;  they  are  those  in  which  n  and  one 
of  the  other  four  quantities  are  unknown. 

From  the  second  formula  it  is  easy  to  obtain  the  value  of  one  of 
the  quantities  a,  q,  I,  and  S,  in  functions  of  the  other  three ;  hence 
the  problem  is  reduced  to  finding  n  by  means  of  the  formula 

Iq 
Now  this  equality  becomes  5-"= — ,    an   equation   of   the   form 

a'^=^b,  a  and  b  being  known  quantities.  Equations  of  this  kind  are 
called  exponential  equations,  to  distinguish  them  from  those  previous, 
ly  considered,  in  which  the  unknown  quantity  is  raised  to  a  power 
denoted  by  a  known  number. 

Before,  however,  we  can  solve  the  exponential  equation  a*=i, 
we  must  understand  the  elementary  properties  of  Continued  Frac- 
tions, which  are  now  to  be  explained. 

Of  Continued  Fractions. 

65 

231.  Having  given  a  fraction  of  the  form  ,    in  which   the 

terms  are  large,  and  prime  with  respect  to  each  other,  we  are  una- 
ble to  discover  its  precise  value,  either  by  inspection  or  by  any  mode 
of  reduction  yet  explained.  The  manner  of  approximating  to  the 
value  of  such  a  fraction  gives  rise  to  a  series  of  numbers,  which 
taken  together,  form  what  is  called  a  contimied  fraction. 

65 

232.  If  we  take,  for  example,  the  fraction  ,  and  divide  both 

its  terms  by  the  numerator  65,  the  value  of  the  fraction  will  not  be 
changed,  and  we  shall  have 

65  _     1 

149  ~  149' 
65 

65        1 
or  effecting  the  division,     T4q"~"2~Xiq 


23 


19 

Now,  if  we  neglect  the  fractional  part   —   of  the  denominator, 

bo 

we  shall  obtain  —  for  the  approximate  value  of  the  given  fraction. 

But  this  value  would  be  too  large,  since  the  denominator  used  was 
too  small. 

19 

If,  on  the  contrary,  mstead  of  neglecting  the  part  — ,   we  were 

bo 

-to  replace  it  by  1,  the  approximate  value  would  be  — ,  which  must 

o 

be  too  small,  since  the  denominator  3  is  too  large.     Hence 

65        1  65        1 

U9-<-2-    ""^    lA9>r 

therefore  the  value  of  the  fraction  is  comprised  between  —  and  — . 

If  we  wish  a  nearer  approximation,  it  is  only  necessary  to  ope- 

19  J.        ,       .        .       .         65 

rr  as  we  did  on  the  given  fraction  -—tt: 
65  °  149 


we  obtain 

19      1 

65      3  +  8 

19' 

hence 

65       1 

149       2  +  1 

3+8 
19' 

8 
If  now,  we  neglect  the  part   — ,  the  denominator  3  will  be  less 

than  the  true  denominator,  and  —  will  be  larger  than  the  number 
3 

which  ought  to  be  added  to  2  ;  hence,  1  divided  by  2+—  will  be 

o 

less  than  the  value  of  the  fraction :  that  is,  if  we  reject  the  frac 


CONTINUED  FRACTIONS.  267 

tional  part  after  the  second  reduction,  we  shall  have 
65        3 
149  "^T* 

If  we  wish  to  approximate  still  nearer  to  the  value  of  the  given 
fraction,  we  find 

8       1 


19      2+£^ 

T» 

and  by  substituting  this  value,  we  have 
65   _  1 
149  ~  2+1 

3  +  1 

2+3 
T 

3 

Now,  if  we  neglect  the  fractional  part  — ,  after  the  third  reduc- 

o 

tion,  we  see  that  2  will  be  less  than  the  real  denominator  ;  hence 

—  will  be  larger  than  the  number  to  be  added  to  3  :  that  is, 

1       7 
3+—=—     is  too  large  ;  hence 


1       2 

is  too  small,  and 

T 

2      16 

2+y=y 

is  too  small ;  therefore 

1       7 
16~16 

is  too  large,  and  hence 

T 

65        7 
149  ^16" 

Now,  as  the  same  train  of  reasoning  may  be  pursued  for  the  re- 
ductions  which  follow,  and  as  all  the  results  are  independent  of  par- 


268  ALGEBRA. 

ticular  numbers,  it  follows  that,  if  we  stop  at  an  odd  reduction,  and 
neglect  the  fractional  part,  the  result  will  he  too  great ;  but  if  we  stop 
at  an  even  reduction,  and  neglect  the  fractional  part,  the  result  xcill 
be  too  small. 

Making  two  more  reductions,  in  the  last  example,  we  have, 
65  _  1 
149  ""2  +1 

3  +  1 

2  +  1 

2  +  1 

1+2. 

2. 

233.  Let  us  take,  as  a  general  case,  the  continued  fraction 

1 

h+i 

c  +  l 

f  d+l 

/+r&c. 

Hence  we  see,  that  a  continued  fraction  has  for  its  mimerator  the 
unit  1,  and  for  its  denominator  a  whole  number,  plus  a  fraction  which 
has  I  for  its  numerator  and  for  its  denominator  a  ivhole  number  plus 
a  fraction,  and  so  on. 

234.  The  fractions 

1  1  1 

a'  a  +  1       a  +  1 

T'  b+1 

c,  &c. 
are  called  approximating  fractions,  because  each  affords,  in  succes- 
sion,  a  nearer  value  of  the  given  fraction. 

1      1      1 

The  fractions  — ,  -7-,  — ,  &c.  are  called  integral  fractions, 
a      0      c  <=.      . 

When  the  continued  fraction  can  be  exactly  expressed  by  a  vulgar 


CONTINUED  FRACTIONS.  269 

fraction,  as  in  the  numerical  examples  already  given,  the  integral  frac- 

1      1      1 

tions  — ,  — ,  — ,  &c.  will  ternninate,  and  we  shall  obtain  an  expres- 
sion for  the  exact  value  of  the  given  fraction  by  taking  them  all. 

235.  We  will  now  explain  the  manner  in  which  any  approximat- 
ing fraction  may  be  found  from  those  which  precede  it. 
1  1 


a 
1 


1 


a 

1st. 

app. 

fraction. 

h 

2d. 

app. 

fraction 

ic+1 

(nhA.'WrJ-, 

-  3d. 

app. 

fraction. 

b  +  l_ 

c 

By  examining  the  third  approximating  fraction,  we  see,  that  its 
numerator  is  formed  by  multiplying  the  numerator  of  the  preceding 
fraction  by  the  denominator  of  the  third  integral  fraction,  and  add- 
ing to  the  product  the  numerator  of  the  first  approximating  frac- 
tion :  and  that  the  denominator  is  formed  by  multiplying  the  deno- 
minator of  the  last  fraction  by  the  denominator  of  the  third  integral 
fraction, and  adding  to  the  product  the  denominator  of  the  first  ap- 
proximating fraction. 

We  should  infer,  from  analogy,  that  this  law  of  formation  is  ge- 

P     Q     R 

neral.     But  to  prove  it  rigorously,  let   — ,  — ,  — ,  be  the  three 

1^      ii      K 

approximating  fractions  for  which  the  law  is  already  established. 

Since  c  is  the  denominator  of  the  last  integral  fraction,  we  have 

from  what  has  already  been  proved 

R       Qc+P 

R'~  Q'c+P'  ' 


1 

action    • 

23* 


Let  us  now  add  a  new  integral  fraction    —   to  those  already  re- 


370  ALGEBRA. 

o 

duced,  and  suppose    —    to  express  the  4th  approximating  fraction. 


S' 

-,.,    ivill . .      ,, 

R'  ^' 


R  S 

It  is  plain  that   jp   will  becoi:'r    -7    by  simply  substituting  for  c. 


c^ — —:  hence, 
a 


Q{c+^)+P 


S  _^V^  d/^^  _{Qc+P)d+Q  _Rd-\-Q 
^'~Q'(c+-)+P~  (^'^+^')rf+Q'  ~  K'd+Q' 

Hence  we  see  that  the  fourth  approximating  fraction  is  deduced 
from  the  two  immediately  preceding  it,  in  the  same  way  that  the 
third  was  deduced  from  the  first  and  second  ;  and  as  any  fraction 
may  be  deduced  from  the  two  immediately  preceded  in  a  similar 
manner,  we  conclude,  that,  the  numerator  of  the  n""  approximating 
fraction  is  formed  by  multiplying  the  numerator  of  the  preceding  frac- 
tion by  the  denominator  of  the  n""  integral  fraction,  and  adding  to  the 
product  the  numerator  of  the  n  —  2  fraction  ;  and  the  denominator  is 
formed  according  to  the  same  law,  from  the  two  preceding  denomina- 
tors. 

236.  If  we  take  the  difference  between  any  two  of  the  consecu. 
tive  approximating  fractions,  we  shall  find,  after  reducing  them  to  a 
common  denominator,  that  the  difference  of  their  numerators  will  be 
equal  to  ±1 ;  and  the  denominator  of  this  difference  will  be  the 
product  of  the  denominators  of  the  fractions. 

1  h 

Taking,  for  example,  the  consecutive  fractions  — ,  and  — 5 , 

we  have, 

1  h  ab-\-\—db  +1 


And 


a       ab-\-\  a{ab+l)         a(a*+l)' 

h  hc+\  -1 


06  +  1       (a*+l)c-fa     (ai-|-l)((ai-f  l)c+a)' 


CONTINUED  FRACTIONS.  271 

m  P     Q     Ji 

To  prove  this  property  in  a  general  manner,  let    pT'  Tv'  "p^'  ^^ 

three  consecutive  approximating  fractions.     Then 
P      Q       PQ'-P'Q 


But 


P'      Q'  P'Q' 

Q      R       R'Q-RQ' 


Q'     R'  Q'R' 

But  R=Qc-irP  and  R'=Q'c+P'  (Art.  235). 

Substituting  these  values  in  the  last  equation,  we  have 

Q     R'  R'Q 


or  reducing 


Q     22       P'Q-PQ! 


p 

Q 

P' 

Q' 

Q 

R' 

Q' 

R' 

Q'     R'  R'Q' 

From  which  we  see  that  the  numerator  of  the  difference 

is  equal,  with  a  contrary  sign,  to  that  of  the  difference 

That  is,  the  difference  hettoeen  the  nmnerators  of  any  tivo  consecutive 
approximating  fractions,  when  reduced  to  a  common  denominator,  is 
the  same  with  a  contrary  sign,  as  that  which  exists  between  the  last 
numerator  and  the  numerator  of  the  fraction  immediately  following. 

But  we  have  already  seen  that  the  difference  of  the  numerators 
of  the  1st  and  2d  fractions  is  equal  to  +1 ;  also  that  the  difference 
between  the  numerators  of  the  2d  and  3d  fractions  is  equal  to  —  1  ; 
hence  the  difference  between  the  numerators  of  the  3d  and  4th  is 
equal  to  +1 ;  and  so  on  for  the  following  fractions. 

Since  the  odd  approximating  fractions  are  all  greater  than  the 
true  value  of  the  continued  fraction,  and  the  even  ones  all  less  (Art. 
232),  it  follows,  that  when  a  fraction  of  an  even  order  is  subtracted 
from  one  of  an  odd  order,  the  difference  should  have  a  plus  sign  ; 
and  on  the  contrary,  it  ought  to  have  a  minus  sign,  when  one  of 
an  odd  order  is  subtracted  from  one  of  an  even. 


272  ALGEBRA. 

237.  It  has  already  been  shown  (Art.  232),  that  each  of  the  ap^ 
proximating  fractions  corresponding  to  the  odd  numbers,  exceeds  the 
true  value  of  the  continued  fraction  ;  while  each  of  those  corres- 
ponding to  the  even  numbers  is  less  than  it.  Hence,  the  difference 
between  any  two  consecutive  fractions  is  greater  than  the  difference 
between  either  of  them  and  the  true  value  of  the  continued  frac- 
tion.  Therefore,  stopping  at  the  n"'  fraction,  the  result  will  be  true 
to  within  1  divided  by  the  denominator  of  the  n""  fraction,  multipli- 
ed  by  the  denominator  of  the  fraction  which  follows.  Thus,  if  Q 
and  R  are  the  denominators  of  consecutive  fractions,  and  we  stop 
at  the  fraction  whose  denominator  is  Q',  the  result  will  be  true  to 

within   ^QTgT*     But  since  a,  b,  c,  d,  &:c.  are  entire  numbers,  the  de- 
nominator R'  will  be  greater  than  Q',  and  we  shall  have 
1  1 

hence,  if  the  result  be  true  to  within  yp^  it  will  certainly  be  true 
to  within  less  than  the  larger  quantity 
1 

that  is,  the  approximate  result  which  is  ohtained,  is  true  to  within 
unity  divided  by  the  square  of  the  denoniinator  of  the  last  approxi- 
mating fraction  that  is  employed. 

829 
If  we  take  the  fraction    ^,„    we  have 
347 

829  _        1 

"347""^"*"  2  +  1 

T+i 

1+2 

3+J_ 
19* 

Here  we  have  in  the  quotient  the  whole  number  2,  which  may 


EXPONENTIAL  QUANTITIES.  273 

either  be  set  aside  and  added  to  the  fractional  part  after  its  value 
shall  have  been  found,  or  we  may  place  1  under  it  for  a  denomina- 
tor  and  treat  it  as  an  approximating  fraction. 

Of  Exponential  Quantities. 

Resolution  of  the  Equation  a'^=b 

238.  The  object  of  this  question  is,  to  find  the  exponent  of  the 
power  to  which  it  is  necessary  to  raise  a  given  number  a,  in  order 
to  produce  another  given  number  b. 

Suppose  it  is  required  to  resolve  the  equation  2^=64.  By  rais- 
ing  2  to  its  different  powers,  we  find  that  2^=64  ;  hence  x=6  will 
satisfy  the  conditions  of  the  equation. 

Again,  let  there  be  the  equation  3-^=: 243.  The  solution  is  x=5. 
In  fact,  so  long  as  the  second  member  Z*  is  a  perfect  power  of  the 
given  number  a,  x  will  be  an  entire  number  which  may  be  obtained 
by  raising  a  to  its  successive  powers,  commencing  at  the  first. 

Suppose  it  were  required  to  resolve  the  equation  2^=6.  By 
making  x—'2,  and  .'c=3,  we  find  2^=4  and  2^=8 :  from  which  we 
perceive  that  x  has  a  value  comprised  between  2  and  3. 

Suppose  then,  that  a;=2-| — y,  in  which  case  x'>l. 

Substituting  this  value  in  the  proposed  equation,  it  becomes, 
_i_  _i_  J.      3 

2^"*'^'=6     or     2^x2  ^'=6;     hence     2^'=—, 

or   (— )   =2,  by  changing  the  members,  and  raising  both  to  the 

x'  power. 

To   determine   a;',  make   successively   x'=^l    and    2;    we    find 

/  3  \'       3  /3  v^      9 

I — j    =—    less  than  2,  and    (— j  =— ,.  which  is  greater  than  2  ; 

therefore  x'  is  comprised  between  1  and  2. 

1 

Suppose     x'=l-\-j;,    inwhicha:>l. 


274  ALGEBRA. 

By  substituting  this  value  in  the  equation    (—1  =2 
/3\,4.-L  3      /3\-l 

/  4  \ --"      3 

^-g-j    =-2      by  reducing. 


The  two  hypotheses  x"=l  and  x"=2,  give  —  which  is  less  than 

o 

3  /4\^     16  7  3 

— ,  and    (  —  1  =— =1+—  which  is  greater  than  — ;    therefore 

x"  is  comprised  between  1  and  2. 
Let  x"=l-\ ;;— ,  there  will  result 

X 

/4\,+-L      3  4       /4\-i-      3 

/9\^"'      4  ,     . 

whence  I— j     =—     by  reducmg. 

Making  successively  x"'=l,  2,  3,  we  find  for  the  two  last  hypo. 

/9\2     81           17                             1          , 
theses  ^—j  =—=1+—,  which  is  <l+y,  and 

(9  \^      729  217  1 

— )  =——-==1+———,  which  is  >1+— :  therefore  x'"  is  com- 
8 /        512  512  o 

prised  between  2  and  3. 

Let  x"'=2-\-—,  the  equation  involving  x'"  becomes 


2+- 


/  y  \        :r'V        4  81  /  9  \  xiv        4 

Kq)        ^IT'^^eiU)     ="3 


/  256  \  x'v      9 
and  consequently  (  g.o  )      ~~q' 

Operating  upon  this  exponential  equation  in  the  same  manner 
as  upon  the  preceding  equations,  we  shall  find   two  entire  num. 


EXPONENTIAL  QUANTITIES.  276 

bera  k  and  &+1,  between  which  a;''  will  be  comprised.     Making 

0^=44 — -,  xv  can  be  determined  in  the  same  manner  as  x^^,  and 
x^ 

so  on. 

Making  the  necessary  substitutions  in  the  equations 

x=2  +  i    a^'=l+^,  a."=l+4r'  ^'"=2-f  ^ 

we  obtain  the  value  of  x  under  the  ibrm  of  a  continued  fraction 

1 

x=2+ J- 

Hence  we  find  the  first  three  approximating  fractions  to  be. 

L    L    1. 

1  '      2  '      5"' 
and  the  fourth  is  equal  to 

3x24-1       7    ^, 
-5^m:2-=12  (^^^-  235), 

which  is  the  value  of  the  fractional  part  to  within 
(12f   ""'  "lii"  (^'^-  2^^)- 

Therefore  ^=2+72=^7^    to  within    -rrr,  and  if  a  greater  de- 

gree  of  exactness  is  required,  we  must  take  a  greater  number  of 
integral  fractions. 

EXAMPLES. 

3»  =  15 X  =        2,46  to  within  0,01. 

10*  =    3 X  =        0,477 0,001. 

2 
5'  =  — x=-    0,25 0,01. 


276  ALGEBRA. 

Theory  of  Logarithms. 
239.  If  we  suppose  a  to  preserve  the  same  value  in  the  equation 

and  y  to  be  replaced  by  all  possible  positive  numbers,  it  is  plain  that 
X  will  undergo  changes  corresponding  to  those  made  in  y.  Now, 
by  the  method  explained  in  the  last  Article,  we  can  determine  for 
each  value  of  y,  the  corresponding  value  of  x,  either  exactly  or  ap- 
proximatively. 

First  suppose  ay.t 
Making  in  succession       x  =0,     1,    2,    3,    4,    5  ,  .  .  .  &c. 
there  will  result  y=a'=l,     a,    a',    a',    a*,   a*,  .  .  .  &c. 

hence,  every  value  of  y  greater  than  unity,  is  produced  by  the  pow' 
ers  of  a,  the  exponents  of  which  are  positive  nuvibers,  entire  or  frac- 
tional ;  and  the  values  of  y  increase  with  x. 

Make  now  x  =0,  —1,  —2,  —3,  —4,  —5,  .  .  .  &;c. 

11111 

there  will  result    y=a'=l,     — ,     — ;,     — ^,     —,,     — r, .  .  .  &c. 
a       a'       a^       a*       a* 

hence,  every  value  of  y  less  than  unity,  is  produced  by  the  powers  of 
a,  of  which  the  exponents  are  negative ;  and  the  value  of  y  dimin' 
ishes  as  the  value  of  x  increases  negatively. 

Suppose  a<l  or  equal  to  the  proper  fraction  — . 

Making x=0,       1,       2,       3,       4,  .  .  .  &c. 

/Ix"  1111 

wefina.     .     .       y=[-^)  =h    -7,    ^.,     ^,     ^,  •  •  •  &c. 

Making x=0,  -1,  -2,  -3,  -4, 

/  1  \° 
we  obtain    .     .       y=(— j  =1,      a',     a'^     a'^     a'\  .  .  .  6cc. 

That  is,  in  the  hypothesis  a<l,  all  numbers  are  formed  with 


THEORY  OF  LOGARITHMS.  277 

the  different  powers  of  «,  in  the  inverse  order  of  that  in  which  they, 
are  formed  when  we  suppose  a>l. 

Hence,  every  possible  positive  number  can  be  formed  with  any  con- 
stant positive  number  whatever,  by  raising  it  to  suitable  powers. 

Remark.  The  number  a  must  always  be  different  from  unity, 
because  all  the  powers  of  1  are  equal  to  1. 

240.  By  conceiving  that  a  table  has  been  formed,  containing  in 
one  column,  every  entire  number,  and  in  another,  the  exponents  of 
the  powers  to  which  it  is  necessary  to  raise  an  invariable  number,  to 
form  all  these  numbers,  an  idea  will  be  had  of  a  table  of  logarithms. 
Hence, 

The  logarithm  of  a  number,  is  the  exponent  of  the  potver  to  which 
it  is  necessary  to  raise  a  certain  invariable  number,  in  order  to  pro- 
duce the  first  number. 

Any  number,  except  1,  may  be  taken  for  the  invariable  number  ; 
but  when  once  chosen,  it  must  remain  the  same  for  the  formation  of 
all  numbers,  and  it  is  called  the  base  of  the  system  of  logarithms. 

Whatever  the  base  of  the  system  may  be,  its  logarithm  is  unity, 
and  the  logarithm  of  1  is  0. 

For,  let  a  be  the  base :  then 

1st,  we  have  a^=a,  whence  log  a=l.     ' 
2d,  «"=!,  whence  log  1=0. 

The  word  logarithm  is  commonly  denoted  by  the  first  three  letters 
log,  or  simply  by  the  first  letter  Z. 

We  will  now  show  some  of  the  advantages  of  tables  of  logarithms 
in  making  numerical  calculations. 

Multiplication  and  Division. 

241.  Let  a  be  the  base  of  a  system  of  logarithms,  and  suppose 
the  table  to  be  calculated.  Let  it  be  required  to  multiply  together 
a  series  of  numbers  by  means  of  their  logarithms.  Denote  the  numw 
bers  by  y,  y',  y",  y'"  .  .  .  &;c.,  and  their  corresponding  logarithms 

24 


278  ALGEBRA. 

by  X,  sf,  x",  x'",  &c.     Then  by  definition  (Art.  240),  we  have 
a'=y,     a"=y',     a"'=3/",     a""=y["  .  .  .  &c. 
Multiplying  these  equations  together,  member  by  member,  and 
applying  the  rule  for  the  exponents,  we  have 

^r+x'+^//+x'//  ,  ,  ,  =y  y'y'y'" or 

x-\-x' +x" -\-x"'  .  .  .  =log?/+  log  3/'+  logi/"+  log 3/'"  .  .  . 
=  log.  yy'y"y"', 
that  is,  the  sum  of  the  logarithms  of  any  number  of  factors  is  equal 
to  the  logarithm  of  the  product  of  those  factors. 

242.  Suppose  it  were  required  to  divide  one  number  by  another. 
Let  y  and  y'  denote  the  numbers,  and  x  and  x'  their  logarithms. 
We  have  the  equations 

a"'— 2/     and     a'''=y' ; 

y 

hence  by  division  a'-^'=-— , 

y 

y 

or  x—x'=:i  log  y—  log  y'=  log  -;-, 

that  is,  the  difference  between  the  logarithm  of  the  dividend  and  the 
logarithm  of  the  divisor,  is  equal  to  the  logarithm  of  the  quotient. 

Consequences  of  these  properties.  A  multiplication  can  be  per- 
formed by  taking  the  logarithms  of  the  two  factors  from  the  tables, 
and  adding  them  together ;  this  will  give  the  logarithm  of  the  pro- 
duct.  Then  finding  this  new  logarithm  in  the  tables,  and  taking 
the  number  which  cortesponds  to  it,  we  shall  obtain  the  required  pro- 
duct. Therefore,  by  a  simple  addition,  we  find  the  result  of  a  7nid- 
tiplication. 

In  like  manner,  when  one  number  is  to  be  divided  by  another, 
subtract  the  logarithm  of  the  divisor  from  that  of  the  dividend,  then 
find  the  number  corresponding  to  this  difterence  ;  this  will  be  the 
required  quotient.  Therefore,  by  a  simple  subtraction,  we  obtain  the 
quotient  of  a  division. 


THEORY  OF  LOGARITHMS.  279 

Formation  of  Powers  and  Extraction  of  Roots. 

243.  Let  it  be  required  to  raise  a  number  y  to  any  power  de- 

Ttl 

noted  by   — .     If  a  denotes  the  base  of  the  system,  and  x  the  loga- 

rithm  of  y,  we  shall  have 

a'=y     or    y=a', 

m 
whence,  raising  both  members  to  the  power    — , 

y'^=a^\ 

—      m  m 

Therefore,  logy"— — .  x= — .log?/, 

that  is,  if  the  logarithm  of  any  number  he  multiplied  by  the  exponent, 
of  the  power  to  which  the  number  is  to  be  raised,  the  product  will  be 
equal  to  the  logarithm  of  that  power. 

As  a  particular  case,  take   n=l  ;  there  will  result    »z.  log  y=: 
log  3/"' ;  an  equation  which  is  susceptible  of  the  above  enunciation. 

244.  Suppose,  in  the  first  equation,  m=l ;  there  will  result 
1 


log  y=  log  y"z=  log  V 


that  is,  the  logarithm  of  any  root  of  a  number  is  obtained  by  divid- 
ing the  logarithm  of  the  number  by  the  index  of  the  root. 

Consequence.  To  form  any  power  of  a  number,  take  the  loga- 
rithm of  this  number  from  the  tables,  multiply  it  by  the  exponent 
of  the  power ;  then  the  number  corresponding  to  this  product  will 
be  the  required  power. 

In  like  manner,  to  extract  the  root  of  a  number,  divide  the  loga- 
rithm of  the  proposed  number  by  the  index  of  the  root,  then  the 
number  corresponding  to  the  quotient  will  be  the  required  root. 
Therefore,  by  a  simple  multiplication,  we  can  raise  a  quantity  to  a 
power,  and  extract  its  root  by  a  simple  division. 


280  ALGEBRA. 

245.  The  properties  just  demonstrated  are  independent  of  any 
system  of  logarithms ;  but  the  consequences  which  have  been  de- 
duced from  them,  that  is,  the  use  that  may  be  made  of  them  in  nu- 
merical calculations,  supposes  the  construction  of  a  table,  contain- 
ing  all  the  numbers  in  one  column,  and  the  logarithms  of  these  num. 
bers  in  another,  calculated  from  a  given  base.  Now,  in  calculating 
this  table,  it  is  necessary,  in  considering  the  equation  a'=y,  to  make 
y  pass  through  all  possible  states  of  magnitude,  and  determine  the 
value  of  a;  corresponding  to  each  of  the  values  of  ?/,  by  the  method 
of  Art.  238. 

The  tables  in  common  use,  are  those  of  which  the  base  is  10 
and  their  construction  is  reduced  to  the  resolution  of  the  equation 
10*=?/.  Making  in  this  equation,  y  successively  equal  to  the  series 
of  natural  numbers,  1,  2,  3,  4,  5,  6,  7  .  .  .,  we  have  to  resolve  the 
equations 

10^=1,     10^=2,     10^=3,     10^=4  .  .  . 

We  will  moreover  observe,  that  it  is  only  necessary  to  calculate 
directly,  by  the  method  of  Art.  238,  the  logarithms  of  the  pi'ime 
numbers  1,  2,  3,  5,  7,  11,  13,  17  ... ;  for  as  all  the  other  entire 
numbers  result  from  the  multiplication  of  these  factors,  their  loga- 
rithms may  be  obtained  by  the  addition  of  the  logarithms  of  the 
prime  numbers  (Art.  241). 

Thus,  since  6  can  be  decomposed  into  2x3,  we  have 
log  6=  log  2+  log  3  ; 
in  like  manner,  24=2='x3 ;  hence  log  24=3  log  2+  log  3. 
Again,  360=23 xS^'X 5  ;  hence 

log  360=3  log  2+2  log  3+  log  5. 

It  is  only  necessary  to  place  the  logarithms  of  the  entire  num. 
bers  in  the  tables ;  for,  by  the  property  of  division  (Art.  242),  we 
obtain  the  logarithm  of  a  fraction  by  subtracting  the  logarithm  of 
the  divisor  from  that  of  the  dividend. 


THEORY  OF  LOGARITHMS.  281 

246.  Resuming  the  equation  10''=y,  if  we  make 


x=0,      1, 

2,    3, 

4, 

5,   .. 

.    n-1, 

n. 

we  have 

y=l,    10, 

100,  1000, 

10000, 

100000, .  . 

10-', 

10". 

And  making 

x=0,   -1, 

-2,  -3, 

-4, 

-5,  .  . 

.  -(n-1), 

— n. 

we  have 

V      1    ^ 

1    1 
100 '  1000' 

1 

1 

1 

10"-" 

1 

y-^^  10' 

10000' 

100000'  • ' 

10"  • 

From  which  we  see  that,  the  logarithm  of  a  whole  numher  will  3e« 
come  the  logarithm  of  a  corresponding  decimal  by  changing  its  sign 
from  plus  to  minus. 

247.  Resume  the  equation  a''=y,  in  which  we  will  first  suppose 

a>l. 
Then,  if  we  make  y=  1  we  shall  have 

a''=l. 
If  we  make  2/<l  we  shall  have 

1 

a-'=y     or     ~=r/<l. 

If  now,  y  diminishes  x  will  increase,  and  when  y  becomes  0,  we 

have  a-'''=-j=0  or  a''=  ao  (Art.  112)  ;  but  no  finite  power  of  a 

is  infinite,  hence  x  =  cd  :  and  therefore,  the  logarithm  of  0  in  a  sys- 
tern  of  ivhich  the  base  is  greater  than  unity,  is  an  infinite  number  and 
negative. 

248.  Again  take  the  equation  a^'^y,  and  suppose  the  base  a<l. 
Then  making,  as  before,  3/=l,  we  have  a''=l. 

If  we  make  y  less  than  1  we  shall  have 

a'=y<\. 
Now,  if  we  diminish  y,  x  will  increase ;  for  smce  a<l  its  powers 
will  diminish  as  the  exponent  x  increases,  and  when  3f=0,  a;  must 
24* 


!S2^!3  ALGEBRA. 

be  infinite,  for  no  finite  power  of  a  fraction  is  0.  Hence,  the  loga- 
rithm of  0  ill  a  system  of  which  the  base  is  less  than  unity,  is  an  in- 
finite  number,  and  positive. 

Logarithmic  and  Exponential  Series. 

349.  The  method  of  resolving  the  equation  a''=h,  explained  in 
Art.  238,  is  sufficient  to  give  an  idea  of  the  construction  of  loga- 
rithmic tables ;  but  this  method  is  very  laborious  when  we  wish  to 
approximate  very  near  the  value  of  x.  Analysts  have  discovered 
much  more  expeditious  methods  for  constructing  new  tables,  or  for 
verifying  those  already  calculated.  These  methods  consist  in  the 
development  of  logarithms  into  series. 

Taking  again  the  equation  a^'^y,  it  is  proposed  to  develop  the 
logarithm  of  y  into  a  series  involving  the  powers  of  y,  and  co-effi- 
cients  independent  of  y. 

It  is  evident,  that  the  same  number  y  will  have  a  different  loga- 
rithm in  different  systems ;  hence  the  log  y,  will  depend  for  its 
value,  1st.  on  the  value  of  y ;  and  2dly,  on  a,  the  base  of  the  sys- 
tem of  logarithms.  Hence  the  development  must  contain  y,  or  some 
quantity  dependent  on  it,  and  some  quantity  dependent  on  the  base  a. 

To  find  the  form  of  this  development,  we  will  assume 
log  y=A+By+Cy^+Dy''+,  &c., 
in  which  A,  B,  C,  &c.  are  independent  of  y,  and  dependent  on  the 
base  a. 

Now,  if  we  make  2/=0,  the  log  y  becomes  infinite,  and  is  either 
negative  or  positive  according  as  the  base  a  is  greater  or  less  than 
unity  (Arts.  247  &  248).  But  the  second  member  under  this  sup- 
position,  reduces  to  A,  a  finite  number  :  hence  the  development  can- 
not be  made  under  that  form. 

Again,  assume 

log  y=Ay+Bf+Cf+Dy'+,  &c. 

If  we  make  3/=0,  we  have 

log  y=±  (X  =0, 


THEORY  OP  LOGARITHMS.  281 

which  is  absurd,  and  hence  the  development  cannot  be  made  under 
the  last  form.  Hence  we  conclude  that,  the  logarithm  of  a  number 
cannot  be  developed  in  the  powers  of  that  number. 

Let  us  now  place  for  y,  l-\-y,  and  we  shall  have 

log  {l+y)  =  Ay+Bf+Cf+Df+  &c (1), 

making  y=-0,  the  equation  is  reduced  to  log  1=0,  which  does  not 
present  any  absurdity. 

In  order  to  determine  the  co-efficients  A,  B,  C  .  .  .,  we  will  fol- 
low the  process  of  Art.  207.  Substituting  z  for  y,  the  equation 
becomes 

log  {l+z)=zAz+Bz^+Cz=+Dz*+  .  .  ,  (2). 

Subtracting  the  equation  (2)  from  (1),  we  obtain 
\og{l+y)-\og(l+z)=A{y-z)+Biy'-z')  +  C(f-^)-\- . . .  (3). 

The  second  member  of  this  equation  is  divisible  by  y—z  ;  we  will 
see,  if  we  can  by  any  artifice,  put  the  first  under  such  a  form  that  it 
shall  also  be  divisible  by  y—z. 

We  have,  log  (l+y)-  log  (1+^)=  log^=:log(l+|^)  ; 

y — 2 

but  since can  be  regarded  as  a  single  number  u,  we  can  de- 

1  +  z  °  ° 

velop  log  (l+«),  or  log  f  1-f-         ),  in  the  same  manner  as 

log  (l+y),  which  gives 

■o«(>+f=^)  =  -^H-.(f^)Vc.(f=i)V... 

Substituting  this  development  for  log  (1+7/)—  log(l+2)  in  the 
equation  (3),  and  dividing  both  members  by  y—z,  it  becomes 

=A+B{y+z)  +  C{y'+yz-\-z')+  .  .  . 
Since  this  equation,  like  the  preceding,  must  be  verified  by  all 


284  ALGEBRA. 

values  of  y  and  *,  make  y=s,  and  there  will  result 

^-p^=^+2%+3C/+4Z>y'+5£2/*+  .  .  . 

Whence,  clearing  the  fraction,  and  transposing 

0=^+25  I  2/+3C  I  f+AD  I  f+^E  I  i/*+  .  .  . 

Putting  the  co-efficients  of  the  different  powers  of  y  equal  to 
zero,  we  obtain  the  series  of  equations 

A-A=Q,     2B+A=0,     3C+25=0,     4Z>+3C=0  .  .  . ; 
whence 


A 

25           A 

SC          A 

B=-^, 

C=- 

2 

3    ~       3' 

~        4    ~      4 

A=A, 

The  law  of  the  series  is  evident ;  the  co-efficient  of  the  n"^  term 

A 

IS  equal  to  qz — ,  according  as  n  is  even  or  odd ;  hence  we  shall  ob- 

tain  for  the  development  of  log  (\-\-y), 


A         A  .     A 

2^ 


log  {\^y)=Ay-—f^—f-—t 


=^(^-Y+-3-T+-5----)    (^)- 

If  we  substitute  —y  for  y,  we  shall  have 

log(l-2/)=A(-r/-^-^-?^+&c.)     (5). 

Hence,  although  the  logarithm  of  a  number  cannot  be  developed 
in  the  powers  of  that  number,  yet  it  may  be  developed  in  the  powers 
of  a  number  greater  or  less  by  unity. 

By  the  above  method  of  development,  the  co-efficients  B,  C,  D, 
E,  &c.  have  all  been  determined  in  functions  of  A ;  but  the  rela- 
tion between  A  and  the  base  of  the  system  is  yet  undetermined. 

The  number  A  is  called  the  modulus  of  the  system  of  logarithms 
in  which  the  log  (1+y),  or  log  (1— y),  is  taken.     Hence, 


THEORY  OF  LOGARITHMS.  '4St5^ 

The  modulus  of  a  system  of  logarithns  depends  for  its  value  on 
the  base,  and  if  a  certain  function  of  any  number  be  multiplied  by  it, 
the  product  will  be  the  logarithm,  of  that  number  augmented  by  unity. 

250.  If  we  take  the  logarithm  of  1  +^  in  a  new  system,  and  de- 
note  it  by  l'{\-^y),  we  shall  have 

'•(i+J')='i'(i'-y+T-T+T-  *=•)    <«> 

in  which  A'  is  the  modulus  of  the  new  system. 

If  we  suppose  y  to  have  the  same  value  as  in  equation  (4),  we 
shall  have 

\\\+y):\{\+y)::A'  :A, 

for,  since  the  series  in  the  second  members  are  the  same  they  may 
be  omitted.     Therefore, 

The  logarithms  of  the  same  number,  taken  in  two  different  systems, 
are  to  each  other  as  the  moduli  of  those  systems. 

1f)\.  If  we  make  the  modulus  A'—\,  the  system  of  logarithms 
which  results  is  called  the  Naperian  System.  This  was  the  first 
system  known,  and  was  invented  by  Baron  Napier,  a  Scotch  Ma- 
thematician. 

With  this  modification  the  proportion  above  becomes 

V{l+y):\{l+y):'.  1  :  A 
or  A.V{l+y)=\{l+y). 

Hence  we  see  that,  the  Naperian  logarithm  of  any  number,  muL 
tiplied  by  the  modulus  of  another  system,  will  give  for  a  product  the 
logarithm  of  the  same  number  in  that  system. 

252,  Again,     A  .\'{\-^ij)=\{\-\-y)     gives 

That  is,  tlie  logarithm  of  any  number  divided  by  the  modulus  of  the 
system,  is  equal  to  the  Naperian  logarithm  of  the  same  number. 


253.  If  we  take  the  Naperian  logarithm  and  make  y=l,  equa- 
tion  (6)  becomes 

1111 

a  series  which  does  not  converge  rapidly,  and  in  which  it  would  be 
necessary  to  take  a  great  number  of  terms  for  a  near  approxima- 
tion. In  general,  this  series  will  not  serve  for  determining  the  loga- 
rithms  of  entire  numbers,  since  for  every  number  greater  than  2 
we  should  obtain  a  series  in  which  the  terms  would  go  on  increasing 
continually. 

The  following  are  the  principal  transformations  for  converting  the 
above  series  into  converging  series,  for  the  purpose  of  obtaining  the 
logarithms  of  entire  numbers,  which  are  the  only  logarithms  placed 
in  the  tables. 

First  Transformation. 

Taking  the  Naperian  logarithm  in  equation  (6),  making  y= — , 
and  observing  that 

r(l4-— )=!'(! +^)—l'2^»     it  becomes 

l'(l+.)-l'z=i-^+^-^+  &c.     (7). 

This  series  becomes  more  converging  as  z  increases  ;  besides  thfef 
first  member  of  this  equation  expresses  the  difference  between  two 
consecutive  logarithms. 

Making  z=l,  2,  3,  4,  5,  &c,  we  have 
1111 

1111 


THEORY  OF  LOGARITHMS.  367 

1111 

M-r3=---+^j_— + . . . 

,.6-l-4=l-i-+-! L.. 

4      32^  192      1024 

The  first  series  will  give  the  logarithm  of  2  ;  the  second  series 
will  give  the  logarithm  of  3  by  means  of  the  logarithm  of  2 ;  the 
third,  the  logarithm  of  4,  in  functions  of  the  logarithm  of  3  .  .  .  &c. 
The  degree  of  approximation  can  be  estimated,  since  the  series  are 
composed  of  terms  alternately  positive  and  negative  (Art.  203). 

Second  Transformation. 

A  much  more  converging  series  is  obtained  in  the  following  man- 
ner. 

In  the  series 

x"     aP      X* 

r(i+.)=.--+---+... 

substitute  —  x  for  a; ;  and  it  becomes 

ar"      3P      X* 


!'(!-)=     ~      2       3       4       '•• 

Subtracting  the  second  series  from  the  first,  observing  that 

1+x 
r(l4-x)— r(l— a;)=r- ,  we  obtain 

This  series  will  not  converge  very  rapidly  unless  a;  is  a  very 
small  fraction,  in  which  case, will  be  greater  than  unity,  but 

will  differ  very  little  from  it. 

l-\-x  1 

Take =1H ,     2  being  an  entire  number : 

1 — X  z  * 


we  have  (14-x)z=(l— a;)(j;+l):  whence  x-. 


22+1 


299  ALGfBBRA. 

Hence  the  preceding  series  becomes    l'(l+' — )  or 

Vi,+i^-n=i{^+^-  +  g(^+  . . .) 

This  series  also  gives  the  difference  between  two  consecutive 
logarithms,  but  it  converges  much  more  rapidly  than  the  series  (7). 
Making  successively  2=1,  2,  3,  4,  5  .  .  .,  we  find 

/I         1  1  1  \ 

r3--r2=2(i-+3i^3+^+y^+  •  •  •)' 

Let  2=100  ;  there  will  result 

noi=noo+2(-^+^-+^+  . . .) ; 

whence  we  see,  that  knowing  the  logarithm  of  100,  the  first  term 
of  the  series  is  sufficient  for  obtaining  that  of  101  to  seven  places 
of  decimals. 

The  Naperian  logarithm  of  10  may  be  deduced  from  the  third  and 
fourth  of  the  above  equations,  by  simply  adding  the  logarithm  of  2 
to  that  of  5  (Art.  241).  This  number  has  been  calculated  with 
great  exactness,  and  is  2,302585093. 

There  are  formulas  more  converging  than  the  above,  which  serve 
to  obtain  logarithms  in  functions  of  others  already  known,  but  the 
preceding  are  sufficient  to  give  an  idea  of  the  facility  with  which 
tables  may  be  constructed.  We  may  now  suppose  the  Naperian 
logarithms  of  all  numbers  to  be  known. 


THEORY  OP  LOGARITHMS.  »»» 

254.  We  have  already  observed  that  the  base  of  the  common 
system  of  logarithms  is  10  (Art.  245).  We  will  now  find  ita 
modulus. 

\'{l+y)  :  \{l+y)  :  :  1  :  A     (Art.  250). 
If  we  make  y=9,  we  shall  have 

no  :  no  :  :  1  :  ^. 
But  the  1'10  =  2,302585093  .  .  .  and  110  =  1  (Art.  245);  hence 

A= — ——=0,434294482  the  modulus  of  the  common  sys- 

2,302585093         '  '' 

tern. 

If  now,  we  multiply  the  Naperian  logarithms   before  found,  by 

this   modulus,   we   shall    obtain   a   table    of    common    logarithms 

(Art.  251). 

255.  All  that  now  remains  to  be  done  is  to  find  the  base  of  the 
Napeiian  system.  If  we  designate  that  base  by  e,  we  shall  have 
(Art.  250), 

I'e  :  le  :  :  1  :  0,434294482. 
But  1V  =  1  (Art.  240):  hence 

1  :  1  e  :  :  1  :  0,434294482, 
or  16=0,434294482. 

But  as  we  have  already  explained  the  method  of  calculating  the 
common  tables,  we  may  use  them  to  find  the  number  whose  loga- 
rithm is  0,434294482,  which  we  shall  find  to  be  2,718281828  : 
hence 

6=2,718281828. 

We  see  from  the  last  equation  but  one  that,  the  modulas  of  the 
common  system  is  equal  to  the  common  logarithm  of  the  Naperian 
base. 


25 


290 


CHAPTER  VI. 

General  Theory  of  Equations. 

256.  The  most  celebrated  analysts  have  tried  to  resolve  equa- 
tions of  any  degree  whatever,  but  hitherto  their  efforts  have  been 
unsuccessful  with  respect  to  equations  of  a  higher  degree  than  the 
fourth.  However,  their  investigations  on  this  subject  have  conduct- 
ed them  to  some  properties  common  to  equations  of  every  degree, 
which  they  have  since  used,  either  to  resolve  certain  classes  of 
equations,  or  to  reduce  the  resolution  of  a  given  equation  to  that  of 
one  more  simple.  In  this  chapter  it  is  proposed  to  make  known 
these  properties,  and  their  use  in  facilitating  the  resolution  of  equa- 
tions. 

257.  The  development  of  the  properties  relating  to  equations  of 
every  degree,  leads  to  the  consideration  of  polynomials  of  a  parti- 
cular nature,  and  entirely  different  from  those  considered  in  the  first 
chapter.     These  are,  expressions  of  the  form 

Ax"'-fBx''""'+ac'"-=+  .  .  .  +Tt  +  U, 
in  which  m  is  a  positive  whole  number ;  but  the  co-efficients 
A,  B,  C,  .  .  .  T,  U,  denote  any  quantities  whatever,  that  is,  entire 
bv  fractional  quantities,  commensurable  or  incommensurable.  Now, 
in  algebraic  division,  as  explained  in  Chapter  I,  the  object  was  this, 
viz.  :  £iven  two  polynomials  entire,  with  reference  to  all  the  letters 
and  particular  numbers  involved  in  them,  tojind  a  third  polynomial 
of  the  same  kind,  ichich,  mulliplied  by  the  second,  would  produce  t'lc 
first. 

But  when  we  have  two  polynomials, 

Ax'"  +  B.r"-'+Cx'"-='+  .  .  .  +Tx  +  U, 
AV+B'o.-'-'+C'a;''  =+  .  .  .  +T'a'+U  , 

which  are  necessarily  entire  only  with   respect  to  x,  and  in  which 
the  co-efficicnts  A,  B,  C  .  .  .,  A',  R',  C  .  .  .,  may  be  any  quantities 


GKNKIIAL  PliOPilUTIKS  OF  KiiLATIO.\S.  291 

whatever,  it  may  be  proposed  to  find  a  third  polynomial,  of  the  samo 
tbrm  and  nature  as  the  two  preceding,  W«cA  mtdtiplied  hythcsecondy 
will  re-produce  the  first. 

The  process  for  effecting  this  division  is  analogous  to  that  for 
common  division  ;  but  there  is  this  difference,  viz.  :  In  this  last,  the 
first  term  of  each  partial  dividend  must  be  exactly  divisible  by  the 
first  term  of  the  divisor ;  whereas,  in  the  new  kind  of  division,  we 
divide  the  first  term  of  each  partial  dividend,  that  is,  the  part  affect- 
ed with  the  highest  power  of  the  principal  letter,  by  the  first  term  of 
the  divisor,  whether  the  co-efficient  of  the  corresponding  partial 
quotient  is  entire  or  fractional ;  and  the  operation  is  continued  until 
a  quotient  is  obtained,  which,  multiplied  by  the  divisor,  will  cancel  the 
last  partial  dividend,  in  which  case  the  division  is  said  to  be  exact ; 
or,  until  a  remainder  is  obtained,  of  a  degree  less  than  that  of  the 
divisor,  with  reference  to  the  principal  letter,  in  which  case  the  di- 
vision is  considered  impossible,  since  by  continuing  the  operation, 
quotients  would  be  obtained  containing  the  principal  letter  affected 
with  negative  exponents,  or  this  same  letter  in  the  denominator  of 
them,  which  would  be  contrary  to  the  nature  of  the  question,  which 
requires  that  the  quotient  should  be  of  the  same  form  as  the  pro- 
posed polynomials. 

258.  To  distinguish  polynomials  which  are  entire  with  reference 
to  a  letter,  x  for  example,  but  the  co-efficients  of  which  are  any 
quantities  whatever,  from  ordinary  polynomials,  that  is,  from  poly, 
nomials  which  are  entire  with  reference  to  all  the  letters  and  parti, 
cular  numbers  involved  in  them,  it  has  been  agreed  to  call  the  first 
entire  functions  of  x,  and  the  second,  rational  and  entire  polynO' 
mials. 

General  Properties  of  Equations. 

259.  Every  complete  equation  of  the  ?«"'  degree,  m  being  a  po- 
sitive  whole  number,  may,  by  the  transposition  of  terms,  and  by 
the  division  of  both  members  by  the  co-efficient  of  af",  be  put  under 
the  form 


292  ALGEBRA. 

3r-\-?x^-'+Qx'^-^+  .  .  .  +Ta;+U=0; 

P,  Q,  R  .  .  .  T,  U,  being  co-efficients  taken  in  the  most  general  al- 
gebraic sense. 

Any  expression,  whatever  the  nature  of  it  may  be,  that  is,  numeri- 
cal or  algebraic,  real  or  imaginary,  which,  substituted  in  place  of  x 
in  the  equation,  renders  Us  first  member  equal  to  0,  is  called  a  root  of 
this  equation. 

260.  As  every  equation  may  be  considered  as  the  algebraic  trans- 
lation of  the  relations  which  exist  between  the  given  and  unknown 
quantities  of  a  problem,  we  are  naturally  led  to  this  principle,  viz. 
EVERY  EauATiON  hos  at  least  one  root.  Indeed,  the  conditions  of 
the  enunciation  may  be  incompatible,  but  then  we  must  suppose 
that  we  shall  be  warned  of  it  by  some  symbol  of  absurdity,  such  as  a 
formula,  containing  as  a  necessary  operation,  the  extraction  of  an 
even  root  of  a  negative  quantity ;  yet  there  will  still  exist  an  ex- 
pression which,  substituted  for  x  in  the  equation,  will  satisfy  it.  We 
will  admit  this  principle,  which  we  shall  have  occasion  to  verify  here- 
after  for  most  equations. 

The  following  proposition  may  be  regarded  as  the  fundamental 
property  of  the  theory  of  equations. 

First  Property. 

261.  If  a  is  a  root  of  the  equation 

x'"+Pa;"-'+Qx"'--+  .  .  .  Tx  +  U  =  0, 

the  first  member  of  it  is  divisible  by  x—a  ;  and  reciprocally,  if  a 
factor  of  the  form  x—a,  will  divide  the  first  member  of  the  proposed 
equation,  a  is  a  root  of  it. 

For,  perform  the  division,  and  see  what  takes  place  when  the  ope- 
ration is  continued  until  the  exponent  of  x,  in  the  first  term  of  the 
dividend,  becomes  0. 


GENERAL  PROPERTIES  OF  EQUATIONS.  293 

This  operation  is  of  thu  nature  of  that  spoken  of  in  Art.  257, 
since  a,  P,  Q,  .  .  .  are  any  quantities  whatever. 


+  p|  +pl         +Pg  +Va^~' 


+  Pa 


4-T 


By  reflecting  a  little  upon  the  manner  in  which  the  partial  quo- 
tients are  obtained,  we  shall  first  discover  from  analogy,  and  after- 
wards by  a  method  employed  several  times  (Arts.  59  &  127),  a  law 
of  formation  for  the  co-efficients  of  these  quotients ;  and  we  may 
conclude,  1st.  that  there  will  be  m  partial  quotients,  2d.  that  the  co- 
efficient  of  the  m"'  quotient,  that  is  of  x°,  must  be 
a->  +  Pa'"-=^+Qa'"-='+  .  .  .  +T, 

T  being  the  co-efficient  of  the  last  term  but  one  of  the  proposed 
equation. 

Hence,  by  multiplying  the  divisor  by  this  quotient,  and  reducing 
it  with  the  dividend,  we  obtain  for  a  remainder 

a-+Pa--«+Qa"-2+  •  •  •  +Ta  +  U. 

Now,  by  hypothesis  a  is  a  root  of  the  equation  ;  hence,  this  re- 
mainder  is  nothing,  since  it  is  nothing  more  than  the  result  of  the  sub- 
stitution of  a  for  X  in  the  equation  ;  therefore  the  divisio7i  is  exact. 

Reciprocally,  if  x— a  is  an  exact  divisor  of  a;""  +  Pa;'^-~'+  .  .  .,  the 
remainder  a"'+Pa'"^*+  .  .  .  will  be  nothing ;  therefore  (Art.  259), 
a  is  a  root  of  the  equation. 

262.  From  this  it  results  that,  in  order  to  discover  whether  a  bi- 
nomial  of  the  form  x—a  is  an  exact  divisor  of  a  polynomial  involv- 
25* 


294  ALGEBRA. 

ing  X,  it  will  be  sufficient  to  see  it'  the  result  of  the  substitution  of  a 
for  X,  is  equal  to  0. 

To  ascertain  whether  a  is  a  root  of  a  polynomial  involving  x, 
which  is  placed  equal  to  0,  it  wii!  be  sufficient  to  try  the  division  of 
it  by  x—a.  If  it  is  exact,  we  may  be  certain  that  a  is  a  root  of  the 
equation. 

263.  Remark.  By  inspecting  the  quotient  of  the  division  in  Art. 
261,  we  perceive  the  following  law  for  the  co-efficients  :  Each  co. 
efficient  is  obtained  hy  multiplying  that  which  precedes  it  by  the  root 
a,  and  adding  to  the  product  that  co. efficient  of  the  proposed  equation 
which  occupies  the  same  rank  as  that  which  we  wish  to  obtain  in  the 
quotient. 

Thus,  the  co-efficient  of  the  3d  term,  a^+Pa+Q,  is  equal  to 
(a-f  P)a+Q,  or  to  the  product  of  the  preceding  co-efficient  a+P, 
by  the  root  a,  augmented  by  the  co-efficient  Q.  of  the  3d  term  of  the 
proposed  equation. 

The  co-efficient  of  the  4th  term  is 

(a2  +  Pa+Q)a-f-R,     or     a'+Pa^+Qa  +  R. 

This  law  should  be  remembered. 

Second  Propei'ty. 

264.  Every  equation  involving  hut  one  unknown  quantity,  has  as 
many  roots  as  there  are  units  in  the  exponent  of  its  degree,  and  no 
more. 

Let  the  proposed  equation  be 

a;"'  +  Px'''-''+Qa;'-2-|-  .  .  .  -fTa;+U  =  0. 
Since  every  equation  has  at  least  one  root  (Art.  260),  if  we  de 
note  that  root  by  a,  the  first  member  will  be  divisible  by  x—a,  and 
we  shall  have  the  identical  equation 

a;'"+Px"  '+  .  .  .  ={:r-a)  (-r"  ^+?'x^--+  ...)...  (1). 
But  by  supposing 

a-   »4-PV  2-f  ...  =0, 
we  olitaiti  :i!i  tiiiation  whiih  has  at  least  oiic  root. 


GENERAL  PROPERTIES  OF  EQUATIONS.  295 

Denote  this  root  by  b,  we  have  (Art.  261), 

a,.— i  +  I^'x^-^^-  .  .  .  =(x-*)  (x"'--  +  P"a;'"-^+  .  .  .). 

Substituting  the  2d  member  for  its  vtUue,  in  equation  (1),  and  we 
have, 
x"'+Pa;--«+  .  .  .  =(x-a)  {x-l)  {x^  s+P'V^^^  ...)...  (2). 

Reasoning  upon  the  polynomial  sf^ -\-V"'ic'^~^ -\-  ...  as  upon  the 
preceding  polynomial,  we  have 

a;"'-HP"x'"^'+  .  .  •  ={x-c)  (a;---  +  P"'a;'"-''+  .  .  .), 
and  by  substitution 

a;"+Px— '+  . .  .  ={x-a)  (x-b)  (x-c)  (x-=*+  ...)...  (3). 

Observe  that  for  each  indicated  factor  of  the  first  degree  with 
reference  to  x,  the  degree  of  x  in  the  polynomial  is  diminished  by 
unity  ;  therefore,  after  m—2  factors  of  the  first  degree  have  been 
divided  out,  the  exponent  of  x  will  be  reduced  to  m— (wi— 2),  or  2 ; 
that  is,  we  shall  obtain  a  polynomial  of  the  second  degree  with  refe- 
rence to  X,  which  can  be  decomposed  into  the  product  of  two  factors 
of  the  first  degree,  (x—k)  (x—l)  (Art.  142).  Now,  as  the  m—'2 
factors  of  the  first  degree  have  already  been  indicated,  it  follows 
that  we  have  the  identical  equation, 

x-^+Px— >+  .  •  .  =(x-a)  (x-b)  (x-c)  .  .  .  (x-k)  (x-l). 

From  which  we  see,  that  the  Jirst  member  of  the  proposed  equU' 
Hon  is  decomposed  into  m  factors  of  the  first  degree. 

As  there  is  a  root  corresponding  to  each  divisor  of  the  first  de. 
gree  (Art.  261),  it  follows  that  the  m  factors  of  the  first  degree 
X— a,  x—b,  X— c  .  .  ,,  give  the  m  roots  a,  b,  c  .  .  .  for  the  proposed 
equation. 

Hence,  the  equation  can  have  no  other  roots  than  a,  b,  c  .  .  .  k,  I, 
since  if  it  had  a  root  «,  different  from  a,  b,  c  .  .  .  1,  it  would  follow 
that  it  would  have  a  divisor  x— a,  different  from  x—a,  x—b, 
x—c  .  .  .  X— /,  which  is  impossible. 

Finally,  every  equation  of  the  m""  degree  has  m  roots,  and  can 
have  no  more. 


296  ALGEBRA. 

265.  There  are  some  equations  in  which  the  number  of  roots  la 
apparently  less  than  the  number  of  units  in  the  exponent  of  their 
degree.  They  are  those  in  which  the  first  member  is  the  product 
of  equal  factors,  such  as  the  equation 

{x-ay{x-bf{x-cy{x-dy^Q, 

which  has  hut/our  different  roots,  although  it  is  of  the  10th  degree. 

It  is  evident  that  no  quantity  a,  different  from  a,  b,  c,  d,  can  veri- 
fy it ;  for  if  it  had  this  root  a,  the  first  member  would  be  divisible 
by  x—cc,  which  is  impossible. 

But  this  is  no  reason  why  the  proposed  equation  should  not  have 
ten  roots, /our  of  which  are  equal  to  a,  three  equal  to  b,  two  equal 
to  c,  and  one  equal  to  d. 

266.  Consequence  of  the  second  property. 

The  first  member  of  every  equation  of  the  m""  degree,  havmg  m 
divisors  of  tjie  first  degree,  of  the  form 

x—a,  x—b,  x—c,  .  .  ".  x—k,  x—l, 
if  we  multiply  these  divisors  together,  two  and  two,  three  and 
three  .  .  .,  we  shall  obtain  as  many  divisors  of  the  second,  third,  &c., 
degree  with  reference  to  x,  as  we  can  form  different  combinations  of 
m  quantities,  taken  two  and  two,  three  and  three,  &c.  Now  the 
number  of  these  combinations  is  expressed  by 

m—\        7n  —  2 
m.-^-,m.-^...  (Art.  163). 

Thus,  the  proposed  equation  has  m .  — - —  divisors  of  the  se- 

7,n  — 1        7)1  —  2      ,.  .  ^   ,       ,  .   ,  1  1 

cond  degree,  m  .  — - — .  — ^ —  divisors  of  the  third  degree,  and 


80  on. 

Composition  of  Equations. 

267.  If  in  the  identical  equation 

x"+Pi'"-»+  .  .  .  =(x— a)  (x—b)  (x—c)  .  .  .  (x—l), 
we  perform  the  multiplication  of  four  factors,  we  have 


COMPOSITION  OF  EQUATIONS. 


297 


x'-a 

aP+ab 

x'-abc 

x-\-abcd  \ 

-b 

+ac 

-abd 

—  c 

■i-ad 

—  acd 

-d 

+  bc 

-bed 

+  bd 

i 

+cd 

) 

=0. 


If  we  perform  the  multiplication  of  the  m  factors  of  the  second 
member,  and  compare  the  terms  of  tlie  two  members,  we  shall  find 
the  following  relations  between  the  co-efficients  P,  Q,  R,  .  .  .  T,  U, 
and  the  roots  a,  b,  c,  .  .  .  k,  /,  of  the  proposed  equation,  viz. 

-a-b-c  .  .  .  -k-l=^P,  or  a  +  b  +  c+  .  .  .  +A-+/=-P; 
ab-\-ac-\-   .  .  .    +^'/=Q 

—  abc—abd  .  .  .  — zX-Z=R,  or  abc-\-abd -{-ikl^—K  ; 


±abcd  .  .  .  kl^\],  or  abed  .  .  .  /:/=±U. 

The  double  sign  has  been  placed  in  the  last  relation,  because  the 
product  —ax—bx—c  ...  X—l  will  he  plus  or  minus  according 
as  the  degree  of  the  equation  is  even  or  odd. 

Hence,  1st.  The  algebraic  sum  of  the  roots,  taken  with  contrary 
signs,  is  equal  to  the  co-efficient  of  the  second  term  ;  or,  the  alge- 
braic sum  of  the  roots  themselves,  is  equal  to  the  co-efficiciit  of  the 
second  term  taken  with  a  contrary  sign. 

2d7  The  sum  of  the  products  of  the  roots  taken  two  and  two, 
with  their  respective  signs,  is  equal  to  tlie  co-efficient  of  the  third 
term. 

The  sum  of  the  products  of  the  roots  taken  three  and  three  with 
their  signs  changed,  is  equal  to  the  co-efficient  of  the  fourth  term  ;  or 
the  co-efficient  of  the  fourth  term,  taken  with  a  contrary  sign,  is 
equal  to  the  sum  of  the  products  of  the  roots  taken  tliree  and  three; 
and  so  on. 

Finally,  the  product  of  all  the  roots,  is  equal  to  the  last  term  ; 
that  is.,  the  product  of  all  the  roots,  taken  with  their  respective  signs, 


298  ALGKBRA. 

is  equal  to  the  last  term  of  the  equation,  taken  with  its  sign,  ichen 
the  equation  is  of  an  even  degree,  and  with  a  contrary  sign,  when  the 
equation  is  of  an  odd  degree.  If  one  of  the  roots  is  equal  to  0,  the 
absolute  term  will  be  0. 

The  properties  demonstrated  (Art.  142),  with  respect  to  equations 
of  the  second  degree,  are  only  particular  cases  of  the  above.  The 
last  term,  taken  with  its  sign,  is  equal  to  the  product  of  the  roots 
themselves,  because  the  equation  is  of  an  even  degree. 

Remarks  on  the  Greatest  Common  Divisor. 

268.  The  properties  of  the  greatest  common  divisor  of  two  poly- 
nomials, were  explained  in  Arts.  66  &  67.  We  shall  here  offer  a 
kw  remarks  to  serve  as  a  guide  in  determining  it. 

Let  A  be  a  rational  and  entire  polynomial,  supposed  to  be 
arranged  with  reference  to  one  of  the  letters  involved  in  it,  a,  for 
example. 

If  this  polynomial  is  not  absolutely  prime,  that  is,  if  it  can  be  de- 
composed into  rational  and  entire  factors,  it  may  be  regarded  as  the 
product  of  three  principal  factors,  viz. 

1st.  Of  a  monomial  factor  A,,  common  to  all  the  terms  of  A. 
This  factor  is  composed  of  the  greatest  common  divisor  of  all  the 
numerical  co-efficients,  multiplied  by  the  product  of  the  literal  fac 
tors  which  are  common  to  all  the  terms. 

2d.  Of  a  polynomial  factor  A^,  independent  of  o,  which  is  com- 
mon to  all  the  co-efficients  of  the  different  powers  of  a,  in  the  ar- 
ranged  polynomial. 

3d.  Of  a  polynomial  factor  A3,  depending  upon  a,  and  in  which 
the  co-efficients  of  the  different  powers  of  a  are  prime  with  each 
other ;  so  that  we  shall  have 

Azz^AjXA^xAg. 

Sometimes  one  or  both  of  the  factors  A,,  A^  reduce  to  unity, 
but  this  is  the  general  form  of  rational  and  entire  polynomials.     It 


GREATEST    COMMON    DIVISOR.  299 

follows  from  this,  that  when  there  is  a  greatest  common  divisor  of 
two  polynomials  A  and  B,  we  shall  have 
D=D,.D,.D3; 

D,  denoting  the  greatest  monomial  common  factor,  D^  the  greatest 
polynomial  factor  independent  of  a,  and  D3  the  greatest  polynomial 
factor  depending  upon  this  letter. 

In  order  to  obtain  D  ^,  find  the  monomial  factor  Aj  common  to  all 
the  terms  of  A.  This  factor  is  in  general  composed  of  literal  fac- 
tors, which  are  found  by  inspecting  the  terms,  and  of  a  numerical 
co-efficient,  obtained  by  finding  the  greatest  common  divisor  of  the 
numerical  co-efficients  in  A. 

In  the  same  way,  find  the  monomial  B,  common  to  all  the  terms  of 
B;  then  determine  the  greatest  factor  Dj  common  to  Aj  a?uZB,. 

This  factor  Dj,  is  set  aside,  as  forming  the  first  part  of  the  re- 
quired common  divisor.  The  factors  A,  and  Bj  are  also  suppressed 
in  the  proposed  polynomials,  and  the  question  is  reduced  to  finding 
the  greatest  common  divisor  of  two  new  polynomials  A'  ani  B' 
which  do  not  contain  a  common  monomial  factor.  It  is  then  to  be 
understood  that  the  process  developed  below,  is  to  be  applied  to 
these  two  polynomials. 

269.  Several  circumstances  may  occur  as  regards  the  number 
of  letters  that  may  be  contained  in  A'  and  B'. 

\st.   When  A'  and  B'  contain  but  one  letter  a. 

When  A'  and  B'  are  arranged  with  reference  to  a,  the  coetli- 
cients  will  necessarily  be  frime  with  each  other;  therefore  in  this 
case,  we  shall  only  have  to  seek  for  the  greatest  common  factor  de- 
pending upon  a,  viz.  D3. 

In  order  to  obtain  it,  we  must  first  prepare  the  polynomial  of  the 
highest  degree,  so  that  its  first  term  may  be  exactly  divisible  by 
the  first  term  of  the  divisor.  This  preparation  consists  in  midtiply. 
ing  the  whole  dividend  by  the  co-efficient  of  the  first  term  of  the  divi- 
sor, or  by  a  factor  of  this  co-efficienl,  or  by  a  certain  pmver  of  it,  in 


300  ALGEBRA. 

order  that  we  may  be  able  to  execute  several  operations,  without 
any  new  preparations  (Art.  68). 

The  division  is  then  performed,  continuing  the  operation  until  a 
remainder  is  obtained  of  a  lower  degree  than  the  divisor. 

If  there  is  a  factor  common  to  all  the  co-efficients  of  the  remainder, 
it  must  be  suppressed,  as  it  cannot  form  a  part  of  the  required  divi- 
sor ;  after  which,  we  operate  with  the  second  polynomial,  and  this 
remainder,  in  the  same  way  we  did  with  the  polynomials  A'  and  B' . 

Continue  this  series  of  operations  until  a  remainder  is  obtained 
which  will  exactly  divide  the  preceding  remainder,  this  remainder 
will  be  the  greatest  common  divisor  D^  of  A'  and  B' ;  and  D,  XD3 
will  express  the  greatest  common  divisor  of  A  and  B  ;  or,  continue 
the  operation  until  a  remainder  is  obtained  independent  of  a,  that  is, 
a  numerical  remainder,  in  which  case,  the  two  polynomials,  A'  and 
B'  will  be  prime  with  each  other. 

2d.   When  A'  and  B'  contain  two  letters  a  andh. 

After  having  arranged  the  polynomials  with  reference  to  a,  we 
first  find  the  polynomial  factor  \\\\\c\\  is  independent  of  a,  if  there 
is  one. 

To  do  this,  we  determine  the  greatest  common  divisor  A,  of  all 
the  co-efficients  of  the  different  powers  of  a  in  the  polynomial  A'. 
This  common  divisor  is  obtained  by  applying  the  rule  for  finding 
the  greatest  common  divisor  of  several  polynomials,  as  well  as  the 
rule  for  the  last  case,  since  these  co-efficients  contain  only  one  let- 
ter b.  In  the  same  way  we  determine  tlie  greatest  common  divisor  B, 
of  all  the  co-efficients  ofB'.  Then  comparing  A^  and  B^,wcset 
aside  their  greatest  common  divisor  D2,  as  forming  a  part  of  the  re- 
quired greatest  common  divisor ;  and  we  also  suppress  the  factors 
A 2  and  B2,  in  A'  and  B';  which  produces  two  new  polynomials  A" 
(md  B",  the  co-efficients  of  which  are  prime  ivith  each  other,  and  to 
which  we  may  consequently  apply  the  rule  for  the  first  case. 

Care  must  ahoays  he  talen  to  ascertain,  in  each  remainder,  whether 


GREATEST    COMMON    DIVISOR.  301 

the  co-efficients  of  the  different  powers  of  the  letter  a,  do  not  contain  a 
common  factor,  which  jiiust  be  suppressed,  as  not  forming  a  part  of 
the  common  divisor.  We  have  already  seen  that  the  suppression 
of  these  factors  is  absolutely  necessary  (Art.  68). 

We  shall  in  this  way  obtain  the  common  divisor  D2,  of  A"  and  B", 
and  D,  xDg  XD3J  for  the  greatest  common  divisor  of  the  polyno- 
mials A  and  B. 

Remark.  In  applying  the  rule  for  the  first  case  to  A''  and  B", 
we  could  ascertain  when  these  two  polynomials  were  prime  with 
each  other,  from  this  circumstance,  viz :  a  remainder  would  he  oh. 
tained  which  would  he  either  numerical,  or  a  function  ofh,  hut  inde- 
pendent of  a.  The  greatest  common  divisor  of  A  and  B  would  then 
beD^XD^. 

3d.   When  A' and  B' contain  three  letters,  a,  b,  c. 

After  arranging  the  two  polynomials  with  reference  to  a,  we  de- 
termine  the  greatest  common  divisor  independent  of  a,  which  is  done 
by  applying  to  the  co-efficients  of  the  different  powers  of  a,  in  both 
polynomials,  the  process  for  the  second  case,  since  these  polyno- 
mial co-efficients  contain  but  two  letters,  i  and  c. 

The  independent  polynomial  D^  being  thus  obtained,  and  the  fac- 
tor  Ag  and  B^,  which  have  given  it,  being  suppressed  in  A'  and  B', 
there  will  result  two  polynomials  A"  and  B",  having  their  co-effi- 
cients j?n  me  with  each  other,  and  to  which  the  rules  for  the  preced- 
ing cases  may  be  applied,  and  so  on. 

EXAMPLES. 

1.   Let  there  be  the  two  polynomials 

aW—c''d--a-c'  +  c\     and     4a'd—2ac'  +  2c^—4acd. 
The  second  contains  a  monomial  factor  2.     Suppressing  it,  and 
arranging  the  polynomials  with  reference  to  d,  we  have 

{a''-c^)d''-a-c''  +  c*,     and     {2a^-2ac)d-ac''+c^. 


302  ALGEBRA. 

It  is  first  necesssLTy  to  ascertain  whether  there  is  a  common  divi- 
sor independent  of  d. 

By  considering  the  co-efficients  a^—c^,  and  —a'c'+c*,  of  the 
first  polynomial,  it  will  be  seen  that  —a-c'^+c*  can  be  put  under  the 
form  —c^{a-  —  c^) ;  hence  a^—c^  is  a  common  factor  of  the  co-effi- 
cients of  the  first  polynomial.  In  like  manner,  the  co-efficients  of 
the  second,  2a=  — 2ac,  and  —ac^+c^,  can  be  reduced  to  2a(a—c), 
and  —(^{a—c);  therefore  a—c  is  a  common  factor  of  these  co- 
efficients. 

Comparing  the  two  factors  a'^—c'^  and  a—c,  as  this  last  will  di- 
vide  the  first,  it  follows  that  a—c  is  a  common  factor  of  the  propos- 
ed polynomials,  and  it  is  that  part  of  their  greatest  common  divisor 
which  is  independent  of  d. 

Suppressing  a^  —  c^  in  the  first  polynomial,  and  a-c  in  the  second, 
we  obtain  the  two  polynomials  d'^—c^  and  2ad  —  c^,  to  which  the  or- 
dinary process  must  be  applied. 


d=-c^ 

\2ad-c^ 

ia"d-  —  4:a-c- 

2ad  +  c^ 

+  2ac-d—4.a-c^ 

—  4a-c^  +c*. 

Expkination.  After  having  multiplied  the  dividend  by  4a^,  and 
performed  two  consecutive  divisions,  we  obtain  a  remainder 
—  4aV+cS  independent  of  the  letter  d  ;  hence  the  two  polynomials 
d=  — c*,  and  2ad—c^,  are  prime  with  each  other.  Therefore  the 
greatest  common  divisor  of  the  proposed  polynomials  is  a  —  c. 

Again,  taking  the  same  example,  and  arranging  with  reference 
to  a,  it  becomes,  after  suppressing  the  factor  2  in  the  second  poly- 
nomial, 

{d''-c')a'-c=d--\-c\  and  2(/a='-(2aZ  +  c=)a  +  c'. 

It  is  easily  perceived,  that  the  co-efficient  of  the  different  powers 
of  a  in  the  second  polynomial  are  prime  with  each  other.  In  the 
first  polynomial,  the  co-efficient  —  c'd^+c*,  of  the  second  term,  or 


GREATEST  COMMON  DIVISOR.  303 

<jf  a*,  becomes  —  c^((Z^— c^) ;  whence  tP—c^  is  a  common  factor  of 
the  two  co-efficients,  and  since  it  is  not  a  factor  of  the  second  poly, 
nomial,  it  may  be  suppressed  in  the  first,  as  not  forming  a  part  of 
the  common  divisor. 

By  suppressing  this  factor,  and  taking  the  second  polynomial  for 
a  dividend  and  the  first  for  a  divisor,  (in  order  to  avoid  preparation), 
we  have 

1st.  2da'  —  2cd\a-\-c^  \\a^—c- 
-   cA  I     2d~' 


Rem.  .  .  — 2crf|«+2dc2 

or, a—c, 

by  suppressing  the  common  factor  {  —  2cd—c^) ; 
2d.  a2_^2||Q_^ 


+ac  — c^l  a+c 


Explanation.  After  having  performed  the  first  division,  a  re- 
mainder  is  obtained  which  contains  —2cd—c^,  as  a  factor  of  its 
two  co-efficients ;  for  2dc'^+c^=  —  c{  —  2cd—c^).  This  factor  be- 
ing  suppressed,  the  remainder  is  reduced  to  a  —  c,  which  will  exact- 
ly divide  a^—c'^. 

Hence  a—c  is  the  required  greatest  common  divisor. 

270.  There  is  a  remarkable  case,  in  which,  the  greatest  common 
divisor  may  be  obtained  more  easily  than  by  the  general  method  ; 
it  is  when  one  of  the  two  polynomials  contains  a  Utter  whicJi  is  not 
contained  in  the  other. 

In  this  case,  as  it  is  evident  that  the  greatest  common  divisor  is 
independent  of  this  letter,  it  follows  that,  by  arranging  the  polyno- 
mial  which  contains  it,  with  reference  to  this  letter,  the  required 
common  divisor  will  be  the  same  as  thai  which  exists  between  the  co. 
efficients  of  the  different  powers  of  the  principal  letter  and  the  second 
polynomial,  which,  by  hypothesis,  is  independent  of  it. 


304  ALaEBRA. 

By  this  method,  we  are  led  to  determine  the  greatest  common 
divisor  between  three  or  more  polynomials  ;  but  they  will  be  more 
simple  than  the  proposed  polynomials.  It  often  happens,  that  some 
of  the  co-efficients  of  the  arranged  polynomial  are  monomials,  or, 
that  we  may  discover  by  simple  inspection  that  they  are  prime  with 
each  other  ;  and,  in  this  case,  w,e  are  certain  that  the  proposed  po- 
lynomials are  prime  with  each  other. 

Thus,  in  the  example  of  Art.  269,  treated  by  the  first  method, 
after  having  suppressed  the  common  factor  a  —  c,  which  gives  the 
results, 

d?  —  c-  and  2ad  — (t, 
we  know  immediately  that  these  two  polynomials  are  prime  with 
each  other  ;  for,  since  the  letter  a  is  contained  in  the  second  and 
not  in  the  first,  it  follows  from  what  has  just  been  said,  that  the  com- 
mon  divisor  must  divide  the  co-efficients  2d  and  —  c^,  which  is  evi- 
dently  impossible  ;  hence,  &c. 

2.  We  will  apply  this  last  principle  to  the  two  polynomials 

and  ^adq—^lfg-^-l^ad—lfgq. 

Since  q  is  the  only  letter  common  to  the  two  polynomials,  which, 
moreover,  do  not  contain  any  common  monomial  factors,  we  can  ar- 
range them  with  reference  to  this  letter,  and  follow  the  ordinary 
rule.  But  as  h  is  found  in  the  first  polynomial  and  not  in  the  second, 
if  we  arrange  the  first  with  reference  to  h,  which  gives 

(3c(7-K18c)5  +  30mj3  +  5/Hp9', 
the  required  greatest  common  divisor  will  be  the  same  as  that  which 
exists  between  the  second  polynomial  and  the  two  co-efficients 
3c5'-fl8c     and     'A^m'p-\-bm'pq, 

Now  the  first  of  these  co-efficients  can  be  put  under  tlie  form 
3c(«j'  +  6),  and  the  other  becomes  5772^(5' -f-G) ;  hence  ^  +  6  is  a  com- 
mon factor  of  these  co-efficients.  It  will  therefore  be  sufficient  to 
ascertain  whether  7  +  6,  which  is  a  prune  divisor,  is  a  factor  of  the 
second  polynomial. 


GREATEST    COMMON    DIVISOR.  305 

Arranging  this  polynomial  with  reference  to  q,  it  becomes 
(4atZ-7/g)5-42/g+24ad  ; 

as  the  second  part  2\ad—^2fg  is  equal  to  Q{^ad—lfg),  it  follows 
that  this  polynomial  is  divisible  by  q+Q,  and  gives  the  quotient 
4ad—'7fg.  Therefore  5+6  is  the  greatest  common  divisor  of  the 
proposed  polynomials. 

271.  Remark.  It  may  be  ascertamed  that  q+Q  is  an  exact  di- 
visor of  the  polynomial  (4:ad—7fg)q-{-24:ad—A2fg,  by  a  method 
derived  from  the  property  proved  in  Art.  261. 

Make  5'+6=:0  or  q^  —  Q  in  this  polynomial ;  it  becomes 
(4ad-lfg)x-G+24:ad—42fg, 

which  reduces  to  0  ;  hence  5' 4-6  is  a  divisor  of  this  polynomial. 

This  method  may  be  advantageously  employed  in  nearly  all  the 
applications  of  the  process.  It  consists  in  this,  viz.  after  obtaming 
a  remainder  of  the  first  degree  with  reference  to  a,  when  a  is  the 
principal  letter,  7nake  this  remainder  equal  to  0,  and  deduce  tJie  value 
of  a  from  this  equation. 

If  this  value,  substituted  in  the  remainder  of  the  2d  degree,  de. 
stroys  it,  then  the  remainder  of  the  1st  degree,  simplified  Art.  68, 
is  a  common  divisor.  If  the  remainder  of  the  2d  degree  does  not 
reduce  to  0  by  this  substitution,  we  may  conclude  that  there  is  no 
common  divisor  depending  upon  the  principal  letter. 

Farther,  having  obtained  a  remainder  of  the  2d  degree  with 
reference  to  a,  it  is  not  necessary  to  continue  the  operation  any 
farther.     For, 

Decompose  this  polynomial  into  tico  factors  of  the  1st  degree, 
which  is  done  by  placing  it-  equal  to  0,  and  resolving  the  resulting 
equation  of  the  second  degree. 

When  each  of  the  values  of  a  thus  obtained,  substituted  in  the 

remainder  of  the  3d  degree,  destroys  it,  it  is  a  proof  that  the  remain- 

der  of  the  2d  degree,  simpUjiedi,  is  a  common  divisor ;  when  only 

erne  of  the  values  destroys  the  remainder  of  the  3d  degree,  the  com- 

26* 


306  ALGEBRA. 

mon  divisor  is  the  factor  of  the  1st  degree  with  respect  to  a,  which 
corresponds  to  this  value. 

Finally,  when  neither  of  these  values  destroys  the  remainder  of 
the  3d  degree,  we  may  conclude  that  there  is  not  a  common  divisor 
depending  upon  the  letter  a. 

It  is  here  supposed  that  the  two  factors  of  the  1st  degree  with 
reference  to  a,  are  rational,  otherwise  it  would  be  more  simple  to 
perform  the  division  of  the  remainder  of  the  3d  degree  by  that  of 
the  second,  and  when  this  last  division  cannot  be  performed  exactly, 
we  may  be  certain  that  there  is  no  rational  common  divisor,  for  if 
there  was  one,  it  could  only  be  of  the  first  degree  with  respect  to 
a,  and  should  be  found  in  the  remainder  of  the  second  degree,  which 
is  contrary  to  hypothesis. 

3.  Find  the,  greatest  common  divisor  of  the  two  polynomials 

6x5  -  4r*  —  1  lar*— Sar*— 3x- 1 
and  4x^  +  20;'— 18x2+ 3x  _  5 

A71S.     2x'  — 4x^+1- 1. 

4.  Find  the  greatest  common  divisor  of  the  polynomials 

20x0  — 12x^4- 16x*—15x^  +  14a.-2  —  15x  +  4. 
and  15x*-  9x^+47r'-21x  +28. 

A71S.     5x2  — 3x+4. 

5.  Find  the  greatest  common  divisor  of  the  two  polynomials 

5a'Ir+2aW  +  ca^-Sa'b*  +  bca 
and  a'  +  5a'd—a''b''+5a-M. 

Ans.     a'-^-ab. 

Transformation  of  EquatioJis. 

The  transformation  of  an  equation  consists  in  changing  ita 
form  without  affecting  the  equality  of  its  members.  The  object  of 
a  transformation,  is  to  change  an  equation  from  one  form  to  another 
that  is  more  easily  resolved. 


TRANSFORMATION    OF    EQUATIONS. 


SOT 


First  Trai^sformation. 
To  make  the  second  term  disappear  from  an  equation. 

272.  The  difficulty  of  resolving  an  equation  generally  dimi- 
nishes with  the  number  of  terms  involving  the  unknown  quan- 
tity;  thus,  the  equation  3^=p,  gives  immediately  a;=db  V?, 
whilst  the  complete  equation  x^+p.r4-9=0,  requires  preparation 
before  it  can  be  resolved. 

Now,  any  equation  being  given,  it  can  always  be  transformed 
into  another,  in  which  the  second  term  is  wanting. 

For,  let  there  be  the  general  equation 

x"'+Pj;'^-'  +  Qa;"*'=+  .  .  .  +Tx+U=0. 

Suppose  x=M+x',  u  being  unknown,  and  x'  an  indeterminate  quan- 
tity;  by  substituting  u-\-x'  for  x,  we  obtain 

(«4-a;T+P("+^T"'+Q(«+^K"^+  •  •  •  •  +T(w+x')+U=0  ; 
developing  by  the  binomial  formula,  and  arranging  according  to  the 
decreasing  powers  of  u,  we  have 


V=o. 


Since  a/  is  entirely  arbitrary,  we  may  dispose  of  it  in  such  a  way 

P 

that  we  shall  have  mx'+P=0  ;  whence  x'= .  Substituting  this 

m 

value  of  x'  in  the  last  equation,  we  shall  obtam  an  equation  of  the 
foiTTi, 

«'"+Q'm'"-=+R'u'"-'+  .  .  .  +T'u-f  U'=0. 
in  which  the  second  term  is  wanting. 

If  this  equation  was  resolved,  we  could  obtain  the  values  of  a; 


m-1 

r+mx'|t<"-i  +m.— ^x'* 

u"""-}-  . 

.  .  +x"» 

+P             +{m-l)Px' 
+Q 

+  Px""-^ 
+Qx'"-2 

+   .    .    . 

+Tx' 

+u 

308  ALGEBRA. 

corresponding  to  those  ofj^,  by  substituting  each  of  the  values  of  m 

p 
in  the  equation  a;=M+a;',  or  x=:M . 

m 

Whence  we  may  deduce  the  following  general  rule  : 
In  order  to  make  the  second  term  of  an  equation  disappear,  sub- 
stitute for  the  unknown  quantity  a  new  unknown  quantity,  united  with 
the  co.efficient  of  the  second  term,  taken  with  a  contrary  sign,  and  di- 
vided by  the  exponent  of  the  degree  of  the  equation. 

Let  us  apply  the  preceding  rule  to  the  equation  x--{'px=q.     If 

we  take  x=rM  — — ,  it  becomes   (u — —\  +_p^i_i--\— ^.     or,    by 
performingthe  operations^ and  reducing,    u-  —  —  z=q,    this    equation 


V? 


gives  u=±\/   x+?'  consequently  we  obtain  for  the  two  corres. 
ponding  values  of  a-, 


273.  Instead  of  making  the  second  term  disappear,  an  equation 
may  be  required,  which  shall  be  deprived  of  its  third,  fourth,  &c. 
term ;  this  can  be  obtained  by  placing  the  co-efficient  of  m'"-^ 
W^^  .  .  .  equal  to  0.  For  example,  to  make  the  third  term  disap. 
pear,  we  make  in  the  above  transformed  equation 

m-\ 

m -—x'^+{m-l)Vx'+Q.=0;  . 

from  which  we  obtain  two  values  for  x',  which  substituted   in  the 
transformed  equation  reduces  it  to  the  form 

w'"+P'm"-»+R'u"-='+  .  .  .  T'j/+U'=0. 
Beyond  the  third  term  it  will   be  necessary  to  resolve  equations 
of  a  degree  superior  to  the  second,  to  obtain  the  value  of  x:  thus  to 
cause  the  last  term  to  disappear,  it  will  be  necessary  to  resolve  the 
equation 


TRANSFORMATION  OF  EQUATIONS.  309 

x'"+Px""-'+  .  .  .  TaZ+U^O, 
which  is  nothing  more  than  what  the  proposed  equation  becomes 
when  ar'  is  substituted  for  x. 

P 

It  may  happen  that  the  value  x'= which  makes  the  second 

m 

term  disappear,  causes  also  the  disappearance  of  the  third  or  some 

other  term.     For  example,  in  order  that  the  second  and  third  terms 

may  disappear  at  the  same  time,  it  is  necessary  that  the  equation 

P 

X  = should  agree  with. 


m ^— x'2+('n-l)Pa;'+Q=0. 

P    . 

Now  if  in  this  last  equation,  we  replace  x'  by it  becomes 


2 

therefore,  whenever  this  relation  exists  between  the  co-efficients  P 
and  Q,  the  disappearance  of  the  second  term  involves  that  of  the 
third. 

Rejnarks  upon  the  preceding  Transformation.     Formation  of 
derived  Polynomials^ 

274.  The  relation  x^=u-\-x',  of  which  we  have  made  use  in  the 
two  preceding  articles,  indicates  that  the  roots  of  the  transformed 
equations  are  equal  to  those  of  the  proposed,  diminished  or  increased 
by  a  certain  quantity.  Sometimes  this  quantity  is  introduced  in 
the  calculus,  as  an  indeterminate  quantity,  the  value  of  which  is 
afterwards  fixed  in  such  a  manner  as  to  satisfy  a  given  condition ; 
sometimes  it  is  a  particular  number  of  a  given  value,  which  expresses 
a  constant  difference  between  the  roots  of  a  primitive  equation  and 
those  of  another  equation  which  we  wish  to  form. 

In  short,  the  transformation  which  consists  in  substituting  u-\-x' 
for  X,  in  an  equation,  is  of  very  frequent  use  in  the  theory  of  equa- 


310 


ALGEBRA. 


tions.     Now  there  is  a  very  simple  method  of  obtaining,  in  practice, 
the  transformation  which  resuhs  from  this  substitution. 

To  show  this  we  shall  invert  the  order  of  the  terms  in  u-\-x',  that 
is,  for  X  substitute  x'-{-u  in  the  equation 

a;-+Pa;m-i_|.Qx'"-2_^Rx'"-+  .  .  .  Ta.'+U  =  0; 

it  becomes,  by  developing  and  arranging  according  to  the  ascending 
powers  of  u, 


+Px""-i+(?n-l)Px"" 
+Qx''^^-\-{m—2)Qx"' 


m— 1 


m  —  2 
+  (m-l) — - — Px'"'-^ 


«2+  . , .  jr=o 


+  . . .  +  . 

+Ta;'      +T 

+U 

If  we  observe  how  the  co-efficients  of  the  different  powers  of  u 
are  composed,  we  shall  see  that  the  co-efficient  of  ii"  is  nothing  more 
than  what  the  first  member  of  the  proposed  equation  becomes  when 
x'  is  substituted  in  place  of  x ;  we  shall  hereafter  denote  it  by  X'. 

The  co-efficient  of  v}  is  formed  by  means  of  the  preceding,  or 
X',  by  multiplying  each  of  the  terms  of  X'  by  the  exponent  of  x' 
in  this  term,  and  then  diminishing  this  exponent  by  unity  ;  we  shall 
call  this  co-efficient  Y'. 

The  co-efficient  of  u^  is  formed  from  Y'  by  multiplying  each  of 
the  terms  of  Y'  by  the  exponent  of  x'  in  this  term,  dividing  the  pro- 
duct by  2,  and  then  diminishing  the  exponent  by  unity.  By  calling 
Z' 


this  co-efficient 


it  is  evident  that  Z'  is  formed  from  Y'  in  the 


same  manner  that  Y'  is  formed  from  X'. 

In  general,  the  co-efficient  of  any  term  in  the  above  transformed 
equation,  is  formed  from  the  preceding  one,  by  multiplying  each  of 


TRANSFORMATION  OF  EQUATIONS.  311 

its  terms  by  the  exponent  of  x'  in  this  term,  dividing  the  product  by 
the  number  of  co-efficients  preceding  the  one  required,  and  then  di- 
minishing the  exponents  of  a/  by  unity. 

Z'      V 

This  law,  by  wliich  the  co-efficients  X',  Y',  — ,   — -—  are  deriv- 

ed  from  each  other,  is  evidently  entirely  similar  to  that  which 
regulates  the  different  terms  of  the  formula  for  the  binomial 
(Art.  165). 

The  expressions  Y',  Z',  V,  W  .  .  .  are  called  derived  polyno. 
mials  of  X',  because  Z'  is  deduced  or  derived  from  Y',  as  Y'  is  de- 
rived from  X' :  V  is  derived  from  Z',  as  Z'  is  derived  from  Y',  and 
so  on.  Y'  is  called  the  jirsi  derived  'polynomial,  Z'  the  second,  SfC. 
Recollect  that  X'  is  what  the  first  member  of  the  proposed  equa- 
tion becomes,  when  x'  is  substituted  for  x. 

The  co-efficient  of  the  first  term  of  the  proposed  equation  has 
been  supposed  equal  to  unity  ;  when  this  is  not  the  case,  the  law  of 
formation  for  the  co-efiicients  of  the  transformed  equations  is  entirely 
the  same,  and  the  co-efficient  of  u'"  is  equal  to  that  of  x". 

275.  To  show  the  use  of  this  law  in  practice,  let  it  be  required 
to  make  the  co-efficient  of  the  second  term  of  the  following  equa- 
tion disappear. 

X*  — 12x3  +  170,-  — 9x+7  =  0. 

12 

According  to  the  rule  of  Art.  272,  take  x=u+— ,  or  x=3+tt, 

which  will  give  a  transformed  equation  of  the  4th  degree,  and  of 
the  form 

Z'  V 

and  the  operation  is  reduced  to  finding  the  values  of 

Z'        V 
Y'       V      _ 
^'     ^'      2'      8.3- 


Y' 

=  - 

123; 

Z' 

=  — 

37  ; 

V 
2.3 

=  0. 

31S  AXOEBRA. 

Now  it  follows  from  the  preceding  law,  that . 
X'   =       (3)*-12.(3)='  +  17.(3)2-9.(3y  +  7,  OT     X'=-110 
Y'   ^4.(3)='-36.(3f  +  34.(3)'-9,  or       . 

Z' 

—  =6.(3)2-36.(3)»  +  17,  or      .... 

^=4.(3y-12 

Therefore  the  transformed  equation  becomes 
m4_37j^2_^23m- 110  =  0. 
Again,  transform  the  equation 

4x3_5a'2+7x-9=0 

into  another,  the  roots  of  which  exceed  the  roots  of  the  proposed 
equation  by  unity- 
Take  m=:x+1  ;  there  will   result  x—  —  l+u,  which  gives  the 
transformed  equation 

Z' 

X'+Y'u+--m2+4u'=0. 

X'   =   4.  (-If-   5.(-l)^  +  7.(-iy-9,  or 

Y'   =12.(-l)-^-10.(-iy+.7 

Z' 
-=12.(-1)'-   5 

V 

¥72=' 


X' 

=  - 

-25; 

Y' 

; 

29^ 

Z' 

o 

=  - 

-17; 

v 

.3 

= 

/i. 

Therefore  the  transformed  equation  becomes 
4u^-17u2  +  29ti  — 25r=0. 
The  following  examples  mny  serve  the  student  for  exercises  ; 
Make  the  second  term  vanish  from  the  following  equations. 
1st.  x*— 10x^+7ar'+4x— 9=0. 

Ans,     u^-33u='-118jr-152u-73=0. 
2d.  Sar' +15x=  +  25x— 3  =  0. 

152 
Ans.     2u^ r— =0. 


TRANSFORMATION  OF  EQUATIONS.  313 

Transform  the  equation  So;"  —  ISa^^  +  Ta--^— 8a;—  9  =  0  into  another, 
the  roots  of  which  shall  be  less  than  the  roots  of  the  proposed  by 

1 

the  fraction     — . 
o 

65        102 

Ans.     2u'-9u^-^u^—^i 9""=^' 

We  shall  frequently  have  occasion  for  the  law  of  formation  of 
derived  polynomials. 

276.   These  polynomials  have  the  following  remarkable  proper- 
ties. 

Let  X  or  .t^+Pa^^'-^+Qx^-^  .  .  .  =0,  be  the  proposed  equation, 
and  a,  b,  c,  I,  its  m  roots,  we  shall  then  have  (Art.  244), 

a;'"4-Pa;'"-»+  .  .  .  ={x—a)  (x—b)  (x—c)  .  .  .  (x—l). 

Substituting  a;'+M  (or  to  avoid  the  accents),  x+u  in  the  place  of 
X ;  it  becomes, 

(a;+t«)'"+P(a;+i<)™-^+  .  .  .  ={x-^u—a)  {x-\-u  —  b)  .  .  .  ; 
or  changing  the  order  of  the  terms  in  the  second  member,  and  re- 
garding x—a,  x—b,  .  .  .  each  as  a  single  quantity, 
(a;+u)'"+P(a;+M)"'-^  .  .  .  ={u+x-a)  (u+x-b)  .  .  .  (u+x-b). 

Now,  by  performing  the  operations  indicated  in  the  two  members, 
we  shall,  by  the  preceding  Article,  obtain  for  the  first  membei-, 

Z 

X-\-Yu+—-u^+  .  .  .  M'" ; 

X  being  the  first  member  of  the  proposed  equation,  and  Y,  Z  .  .  . 

the  derived  polynomials  of  this  member. 

With  respect  to  the  second  member,  it  follows  from  Art.  247, 
1st.  That  the  part  involving  u°,  or  the  last  term,  is  equal  to  the 

product   (x  —  a)  (x—b)  .  .  .  (x—l)   of  the   factors   of  the  proposed 

equation  ; 

2d.  The  co-efficient  of  u  is  equal  to  the  sum  of  the  products  of 

these  m  factors  taken  rn—l  and  m—1. 
27 


oil  ALGEBRA. 

Sd.  The  co-efficient  of  u~  is  equal  to  the  sum  of  the  products  of 
these  m  factors  taken  m—2  and  m  — 2  ;   and  so  on. 

Moreover,  the  two  members  of  the  last  equation  are  identical ; 
therefore,  the  co-efficieuts  of  the  same  powers  are  equal.     Hence 

X={x-a)  {x-b)  {x-c)  .  .  .  (.r-/), 

which  was  already  known.  Hence  also,  Y,  or  the  first  derived  po. 
lynomial,  is  equal  to  the  sum  of  the  products  of  the  m  factors  of  ih^ 
first  degree  in  the  proposed  equation,  taken  m  — 1  and  m  — 1;  or 
equal  to  the  sum  of  all  the  quotients  that  can  be  obtained  by  dividing 
X  by  each  of  the  m  factors  of  the  first  degree  in  the  proposed  equa- 
tion ;  that  is, 

,,       X  X  X  X 

Y= i 1 +  .  .  . . 

X — a     X — b     x—c  X — / 

Z 

—  or  the  second  derived  polynomial,  divided  by  2,  is  equal  to  the 

sum  of  the  products  of  the  m  factors  of  the  proposed  equation  taken 
m  —  2  and  m  —  2,  or  equal  to  the  sum  of  the  quotients  that  can  be 
obtained  by  dividing  X  by  each  of  the  factors  of  the  second  degree  ; 
that  is, 

Z_  X  X  X 


2       (x—a)  {x—b)       (a:— a)  (x—c)  {x-k)  {x—l)  ' 

and  so  on. 

Second  Transformation. 
To  make  the  denominators  disappear  from  an  equation. 

277.  Having  given  an  equation,  we  can  always  transform  it  into 
another  of  which  the  roots  will  be  equal  to  a  given  multiple  or  sub- 
mult'iple  of  those  of  the  proposed  equation. 

Take  the  equation 

.r^'  +  Px-"  •+Qa;"'  -+  .  .  .  Ta;+U  =  0, 
and  denote  by  y  the  unknown  quantity  of  a  new  equation,  of  which 


TRANSFORMATION  OF  EQUATIONS.  315 

the  roots  are  K  times  greater  than  those  of  the  proposed  equation. 

y 

If  we  take  2/= Kx,  there  will  result  x=-^  ;   whence,   substituting 
and  multiplying  every  term  by  K"',  we  have 

r+PKr-'+QKV"2+RKV"-'+  . . .  +TK-»y+UK'"=0. 
an  equation  of  which  the  co-efficients  are  equal  to  those  of  the  pro- 
posed equation  multiplied  respectively  by  K",  K*,  K^  K^  K*,  &c. 

This  transformation  is  principally  used  to  make  the  denominators 
disappear  from  an  equation,  when  the  co-efficient  of  the  first  term  is 
unity. 

To  fix  the  ideas,  take  the  equation  of  the  4'''  degree 
a  c  e         g 

if  in  this  equation  we  make  a'=— ,  y  being  a  new  unknown  and  K 
an  indeterminate  quantity,  it  becomes 

,       aK  cK-  eK^*         ^K^ 

Now,  there  may  be  two  cases, 

1st.   Where  the  denominators   h,  d,  f  h,  are  prime  with    each 
other  ;  in  this  hypothesis,  as  K  is  altogether  arbitrary,  take  K=bdfh, 
the  product  of  the  denominators,  the  equation  will  then  become 
y^+adfh  .  y^'  +  clrdf-h'' .  y'^+ePdf-h^  .  y+gb^dfVv'^O, 
an  equation  the  co-efficients  of  which  are  entire,  and  that  of  its  first 
term  unity. 

y 

We  have  besides,  the  equation  a;— ,  ,^.,  ,  to  determine  the  values 

bdjh 

of  X  corresponding  to  those  of  y. 

2d.  When  the  denominators  contain  common  factors,  we  shall 

evidently  render  the  co-efficients  entire  by  taking  for  K  the  small- 

est  multiple  of  all  the  denominators.     But   we   can   simplify  this 

still  more,  by  observing,  that  it  is  reduced  to  determining  K  ir 


316 


ALGEBRA. 


such  a  manner  that  K*,   K^,   K^ .  .  .  shall  contain  the  prime  fac- 
tors which  compose  b,  d,f,  h,  raised  to  powers  at  least  equal  to  tl)Ose 
which  are  found  in  the  denominators. 
Thus,  let  the  equation 

^      5  5  7  13 

'  -¥'^'+12'^'-l50-'^'-9000='- 

y 

Take  a;=— ,  it  becomes 

5k  5fr  7P         ISk' 

^— 6-^'+^2-^'-l50-^-9000=^- 
First  make  A:=9000,  which  is  a  multiple  of  all  the  other  deno- 
minators, it  is  clear  that  the  co-efficients  become  whole  numbers. 

But  if  we  decompose  6,  12,  iSO  and  9000  into  their  factors,  we 
find 

6=2X3,     12=2^x3,     150  =  2x3x5^    9000=2='x32x55 ; 
and  by  simply  making  A:=2x3x5,  the  product  of  the  different  sim- 
ple factors,  we  obtain 

^-=22x3"X5^    P=2='x3='^X5^    ifc*=2*x3'x5S 
whence  we  see  that  the  values  of  k,  P,  P,  k*,  contain  the  prime 
factors  of  2,  3,  5,  raised  to  powers  at  least  equal  to  those  which 
enter  in  6,  12,  150  and  9000. 

Hence  the  hypothesis  k—2x^X^  is  sufficient  to  make  the 
denominators  disappear.  Substituting  this  value,  the  equation 
becomes 

5.2.3.5  ^      5.2-..3='.52  ^     7.2^3^.5==       13.2'».3^5^ 
^'~~2r3T^  ■^■"2^:3      ^'        2X5^^ ~   2^3='.5='    ^^' 
which  reduces  to 

?/4_5_5^3^5  3  52^2_7  22.3-.5?/- 13.2.3^5=0  ; 

or  ?/^-25?/^  +  875y'-1260?/-1170  =  0. 

Hence,  we  perceive  the  necessity  of  taking  k  as  small  a  number 
as  possible :  otherwise,  we  should  obtain  a  transformed  equation, 
having  its  co-cfficients  very  great,  as  may  be  seen   by  reducing 


TUANSFORMATION    OF    EQUATIONS.  317 

the  transformed  equation  resulting  from  the  suppo.Miioii  L~9000  in 
the  preceding  equation. 


whence 


EXAMPLES. 


7         11       25  y 


y^-Uf  +  lly—15  =  0 


13        21  32  43  1  y 

y 

whence 

^5_65^4^1590^_3Q720/-928800]/  +  972000=0. 

278.  The  preceding  transformations  are  those  most  frequently 
used  ;  there  are  others  very  useful,  of  which  we  shall  speak  as  they 
present  themselves  ;  they  are  too  simple  to  be  treated  of  separately. 

In  general,  the  problem  of  the  transformation  of  equations  should 
be  considered  as  an  application  of  the  problem  of  elimination  be- 
tween two  equations  of  any  degree  whatevei',  involving  two  un- 
known quantities.  In  fact,  an  equation  being  given,  suppose  that 
we  wiih  to  transform  it  into  another,  of  which  the  roots  have,  with 
those  of  the  proposed  equation,  a  determined  relation. 

Denote  the  proposed  equation  by  F(a;)  =  0,  (enunciated  function 
of  ic  equal  to  zero),  and  the  algebraic  expression  of  the  relation 
which  should  exist  between  a;  and  the  new  unknown  quantity  y,  by 
F'  {x,y)~Q  ;  the  question  is  reduced  to  fiivling,  by  means  of  these 
two  equations,  a  new  equation  involving  y,  which  will  be  the  re- 
quired  equation.  When  the  unknown  quantity  x  is  only  of  the  first 
degree  in  F'(.r,  .v)  =  0,  the  transformed  equation  is  easily  obtained, 
but  if  it  is  raised  to  the  second,  third  .  .  .  power,  we  must  have  re- 
course  to  the  methods  of  elimination. 
27* 


318  ALGEBRA. 


Elimination. 


279.  To  eliminate  between  two  equations  of  any  degree  what- 
ever, involving  two  unknown  quantities,  is  to  obtain,  by  a  series  of 
operations,  performed  on  these  equations,  a  single  equation  which 
contains  hit  one  of  the  unknown  quantities,  and  which  gives  all  the 
values  of  this  unknown  quantity  that  will,  taken  in  connection  with 
the  corresponding  values  of  the  other  unknown  quantity,  satisfy  at 
the  same  time  both  the  given  equations. 

This  new  equation,  which  is  a  function  of  one  of  the  unknown 
quantities,  is  called  the  final  equation,  and  the  values  of  the  unknown 
quantity  found  from  this  equation,  are  called  compatible  values. 

Of  all  the  known  methods  of  elimination,  the  method  of  the  com- 
mon divisor,  is,  in  general,  the  most  expeditious ;  it  is  the  method 
which  we  are  going  to  develop. 

Let  Y{x,  2/)  =  0  and  F'(.r,  ?/)  =  0  be  any  two  equations  whatever, 
or,  more  simply, 

A=0,  B  =  0. 

Suppose  the  final  equation  involving  y  obtained,  and  let  us  try  to 
discover  some  property  of  the  roots  of  this  equation,  which  may 
serve  to  determine  it. 

Let  y—a  be  one  of  the  compatible  values  of  i/ ;  it  is  clear,  that 
since  this  value  satisfies  the  two  equatio.ns,  at  the  same  time  as  a 
certain  value  of  x,  it  is  such,  that  by  substituting  it  in  both  of  the 
equations,  which  will  then  contain  only  x,  the  equations  will  admit  of 
at  least  one  common  value  of  x ;  and  to  this  common  value  there 
will  necessarily  be  a  corresponding  common  divisor  involving  x. 
Art.  262.  This  common  divisor  will  be  of  the  first,  or  a  higher 
degree  with  respect  to  x,  according  as  the  particular  value  of  y=a 
corresponds  to  one  or  more  values  of  x. 

Reciprocally,  every  value  of  y,  ivhich,  substituted  in  the  two  equa- 
tions, gives  a  common  divisor  involving  x,  is  necessarily  a  compatible 
value,  because  it  then  evidently  satisfies  the  two  equations  at  the 


ELIMINATION.  319 

same  time  with  the  value  or  values  of  x  found  from  this  common  di- 
visor when  put  equal  to  0. 

280.  We  will  remark,  that,  before  the  suhiitution,  the  first  mem- 
bers of  the  equations  cannot,  in  general,  have  a  common  divisor,  which 
is  a  function  of  one  or  both  of  the  unknown  quantities. 

In  fact,  let  us  suppose  for  a  moment  that  the  equations  A  =  0, 
B=:0,  are  of  the  form 

A'xD=0,     B'xD=0. 

D  being  a  function  of  x  and  y. 

Making  separately  D  =  0,  we  obtain  a  single  equation  involving 
two  unknown  quantities,  which  can  be  satisfied  with  an  infinite  num. 
her  of  systems  of  values.  Moreover,  every  system  which  renders 
D  equal  to  0,  would  at  the  same  time  cause  A'D,  B'D  to  vanish,  and 
would  consequently  satisfy  the  equations  A  =  0,  B  =  0. 

Thus,  the  hypothesis  of  a  common  divisor  of  the  two  polynomials 
A  and  B,  containing  x  and  y,  would  bring  with  it  as  a  consequence 
that  the  proposed  equations  were  indeterminate.  Therefore,  if  there 
exists  a  common  divisor,  involving  x  and  y,  of  the  two  polynomials 
A  and  B,  the  proposed  equations  will  be  indeterminate,  that  is,  they 
may  be  satisfied  by  an  infinite  number  of  systems  of  values  of  x 
and  y.  Then  there  are  no  data  to  determine  b.  final  equation  in  y, 
since  the  number  of  values  of  y  is  infinite. 

If  the  two  polynomials  A  and  B  were  of  the  form  A'xD,  B'xD, 
D  being  a  function  of  x  only,  we  might  conceive  the  equation  D=0 
resolved  with  reference  to  x,  which  would  give  one  or  more  values 
for  this  unknown.  Each  of  these  values  substituted  in  A'xD=0 
and  B'  xD=0,  at  the  same  time  with  any  arbitrary  value  of  3,',  would 
verify  these  two  equations,  since  D  must  be  nothing,  in  consequence 
of  the  substitution  of  the  value  of  x.  Therefore,  in  this  case,  the 
proposed  equations  would  admit  of  o. finite  numher  of  values  for  x, 
but  of  an  infinite  number  of  values  for  y  ;  then  there  could  not  exist 
a  final  equation  in  y. 

Hence,  when  the  equations  A  =  0,  B  =  0,  arc  determinate,  that  is, 


320  ALGEBRA. 

when  they  only  admit  of  a  limited  number  of  systems  of  values  for 
X  and  y,  their  first  members  cannot  have  &.  function  of  these  unknown 
quantities  for  a  common  divisor,  unless  a  particular  substitution  has 
been  made  for  one  of  them. 

2S1.  From  this  it  is  easy  to  deduce  a  process  for  obtaining  the 
fnal  equation  involving  y. 

Since  the  characteristic  property  of  every  compatible  value  of 
y  is,  that  being  substituted  in  the  first  members  of  the  two  equations, 
it  gives  them  a  commou  divisor  involving  x,  which  they  had  not  be- 
fore,  (unless  the  equations  are  indeterminate,  which  is  contrary  to 
the  supposition),  it  follows,  that  if  to  the  two  proposed  polynomials, 
arranged  with  reference  to  x,  we  apply  the  process  for  the  greatest 
common  divisor,  we  generally  shall  not  find  one;  but,  by  continuing 
the  operation  properly,  we  shall  arrive  at  a  remainder  independent 
of  X,  and  which  is  a  function  of  y,  which,  placed  equal  to  0,  will 
give  the  required  fnal  equation  ;  for  every  value  of  y  found  from 
this  equation,  reduces  to  nothing  the  last  remainder  of  the  operation 
for  finding  the  common  divisor  ;  it  is,  then,  such,  that  substituted  in 
the  preceding  remainder,  it  will  render  this  remainder  a  common  di- 
visor  of  the  first  members  A  and  B.  Therefore,  each  of  the  roots 
of  the  equation  thus  formed  is  a  compatible  value  of  y. 

282.  AdmUting  that  the  final  equation  may  be  completely  re- 
solved, which  would  give  all  the  compatible  values,  it  would  after- 
wards  be  necessary  to  obtain  the  corresponding  values  of  x.  Now 
it  is  evident  that  it  would  be  suthcieiit  for  this,  to  substitute  the  dif- 
ferent  values  of  y  in  the  remainder  preceding  the  last,  put  the  ])oly- 
nomial  involving  x  which  results  from  it  equal  to  0,  and  find  fiom  it 
the  values  of  a;;  for  these  polynomials  are  nothing  more  than  the 
divisors  involving  x,  which  become  common  to  A  and  B. 

But  as  the  final  equation  is  generally  of  a  degree  superior  to  the 
second,  we  cannot  here  explain  the  methods  of  finding  the  values  of 
y.  Indeed,  our  design  was  principally  to  show  that,  two  equations 
of  any  degree  being  given,  we  can,  without  supposing  the  resolution 


EQUAL  ROOTS.  321 

of  any  equation,  arrive  at  another  equation,  containing  only  one  of 
the  unhnoion  quantities  which  enter  into  the  proposed  equations. 

Of  Equal  Roots. 

283.  An  equation  is  said  to  contain  equal  roots,  when  its  first 
member  contains  equal  factors.  When  this  is  the  case,  the  derived 
polynomial,  which  is  the  sum  of  the  products  of  the  m  fectors  taken 
m— 1  andm— 1  (Art.  276),  contains  a  factor  in  its  different  parts, 
which  is  two  or  more  times  a  factor  of  the  proposed  equation. 

Hence,  there  must  he  a  common  divisor  between  the  first  member  of 
the  proposed  equation  and  its  first  derived  polynomial. 

It  remains  to  ascertain  the  manner  in  which  this  common  divisor 
is  composed  of  the  equal  factors. 

284.  Having  given  an  equation,  it  is  required  to  discover  whether 
it  has  equal  roots,  and  to  determine  these  roots  if  possible. 

Let  X  denote  the  first  member  of  the  equation 

.r"'-+Pa;"'-»+Q.r"'--+  .  .  .  +Tx+U  =  0, 
and  suppose  that  it  contains  n  factors  equal  to  x—a,  n'  factors  equal 
to  x  —  b,  n"  factors  equal  to  x  —  c  .  .  .,  and  contains  also  the  simple 
factors  X — p,  x — q,  x — r  .  •  .  ;  so  that  we  may  have 

X  =  (a;  — i7)"(.r  — Z')"'(a;— c)""  .  .  .  {x—p)  {x—q)  (x—r)  .  .  . 

With  respect  to  Y,  or  the  derived  polynomial  of  X,  we  have 
seen  (Art.  276),  that  it  is  the  sum  of  the  quotients  obtained  by  divid- 
ing  X  by  each  of  the  m  factors  of  the  first  degree  in  the  proposed 
equation.     Now.  since  X  contains  n  factors  equal  to  x—a,  we  shall 

X 

have  n  partial  quotients  equal  to ;  the  same  reasoning  applies 

to  each  of  the  general  factors,  x—b,  x  —  c.  .  .  .     Moreover  we  can 
form  but  one  quotient  equal  to 

XXX 
x—p'     x—q'     x—r 
Therefore,  Y  is  necessarily  of  the  form 


322  ALGEBRA. 

nX       n'X       n"X  XXX 

Y= + -+ 4-  . . .  + + + +  . . . 

x—a     x—b     x—c  x—p     x—q     x—r 

From  this  composition  of  the  polynomial  Y,  it  is  plain  that 
(a;-a)"-»,     {x-b)'"-\     (x-c)""-*  .  .  . 
are  factors  common  to  all  its  terms  ;   hence  the  product 

(x-a)''-'x(a;-i)"'-»X(a;— c)""-'  .  .  . 
is  a  common  divisor  of  Y  ;  moreover,  it  is  evident  that  this  product 
will  also  divide  X,  it  is  therefore  a  common  divisor  of  X  and  Y  ;  and 
it  is  their  greatest  common  divisor.  For,  the  prime  factors  of  X 
are  x — a,  x — b,  x — c  .  .  .  and  x—p,  x  —  q,  x — r  .  .  .  ;  now  x — p, 
x—q,  x—r,  cannot  divide  Y,  since  some  one  of  them  will  be  want- 
ing  in  each  of  the  parts  of  Y,  while  it  will  be  a  factor  of  all  the 
other  parts. 

Hence,  the  greatest  common  divisor  of  X  and  Y  is 
D  =  (a;-fl)"-'(a'-Z')  '   '(.r-c)  ""i  .  .  .  ; 
that  is,  the  greatest  co}nmon  divisor  is  composed  of  the  product  of  those 
factors  which  enter  two  or  more  times  in  the  proposed  equation,  each 
raised  to  a  power  less  hy  unity  than  in  the  given  equation. 

285.  From  the  above  we  deduce  the  following  method  : 

To  discover  whether  an  equation  X  =  0  contains  any  equal  roots, 
form  Y  or  the  derived  j^olynomial  of  X  ;  then  seek  for  the  greatest 
common  divisor  between  X  mid  Y  ;  if  one  cannot  be  obtained,  the 
equation  has  no  equal  roots,  or  equal  factors. 

If  we  find  a  common  divisor  D,  and  it  is  of  the  first  degree,  or  of 
the  form  x—h,  make  x—h=0,  whence  x=h  ;  we  may  then  conclude, 
that  the  equation  has  two  roots  equal  to  h,  and  has  but  one  species  of 
equal  roots,  from  which  it  may  be  freed  by  dividing  X  by  {x—hy. 

If  D  is  of  the  second  d(-gi-ee  with  reference  to  .r,  resolve  the  equa. 
Hon  I)=;0;  there  may  b<:!  two  cases;  the  two  roots  will  be  equal, 
or  they  will  be  unequal.  1st.  When  we  find  'D={x—liY,  the  equa. 
Hon  has  three  reots  equal  to  h,  and  has  but  one  species  of  equal  roots, 
from  which  it  can  be  freed  by  dividing  X  by  (x  —  hf ;  2d,  when  D 


EQUAL     ROOTS.  323 

is  of  the  form  (x—h)  (x—h'),  the  proposed  equation  has  two  roots 
equal  to  h,  and  two  equal  to  h',  from  which  it  may  be  freed  by  divi. 
ding  X  by  {x-hf{x-li'f,  or  by  D=^. 

Suppose  now  that  D  is  of  any  degree  whatever;  it  is  necessary, 
in  order  to  know  the  species  of  equal  roots,  and  the  number  of  roots 
of  each  species,  to  resolve  completely  the  equation  D=:0  ;  and  every 
simple  root  of  D  will  be  twice  a  root  of  the  proposed  equation  ;  every 
double  root  of  D  will  be  three  times  a  root  of  the  proposed  equation  ; 
and  so  on. 


EXAMPLES. 

1.  Determine  whether  the  equation 

2x*-12x'^  +  19.r--6x+9=0 
contains  equal  roots. 

We  have  (Art.  274),  for  the  derived  polynomial 

Sx^  — 36a;"-f38a;— 6. 
Now,  seeking  for  the  greatest  common  divisor  of  these  polyno- 
mials, we  find  D=a'— 3=0,  whence  a;=3  ;    hence    the    proposed 
equation  has  two  roots  equal  to  3. 

Dividing  its  first  memb(;r  by  (x  — 3)^,  we  obtain 

1       , 

2x^  +  1=^0;  whence  x—±~  V—  2. 

Thus  the  equation  is  completely  resolved,  and  its  roots  are 

3,  3,   +  TT  ^^-  2  and  -  —  V-  2. 

2.  For  a  second  example  take  a,-^  — 2x''  +  3x^  — 7.i;--f-8.r  — 3=0  ; 
the  first  derived  polynomial  is  Sa;*  — 8x^  +  9x^—1 4a; +8, 
and  the  common  divisor  r'  —  2x-\-\,  or  (x— 1)^, 
hence  the  proposed  equation  has  three  roots  equal  to  1. 

Dividing  its  first  member  by  {x—lf  or  by  3?—^3?-\-^x—l,  the 
quotient  is 


824  ALGEBRA. 

1±  V- 


x^-]-x-]-3  —  0  ;  whence  x-- 


2 

tlius  the  equation  is  completely  resolved. 
3.  For  a  third  example,  take  the  equation 

.r^  +  5x« + 6x'  —  6a;''  —  1 5a;^  —  3x2 + 8x + 4 = 0  ; 
the  derived  polynomial  is 

7a;«  +  30x5 ^ 30^,4 _ 243,3 _  45^^ _ 6x+ 8  ; 

and  the  common  divisor  is 

x^  +  Sx'+ar'  — 3x— 2.^ 
The  equation  x^  +  Sar'+x^— 3x— 2=0  cannot  be  resolved  directly, 
but  by  applying  the  method  of  equal  roots  to  it,  that  is,  by  seeking 
for  a  common  divisor  between  its  first  member  and  its  derived  poly- 
nomial,  4x''+9x^+2x— 3,  we  find  a  common  divisor,  x+1 ;  which 
proves  that  the  square  of  x+1  is  a  factor  of  x*  +  3x^+x2— 3x— 2, 
and  the  cube  of  x+1,  a  factor  of  the  first  member  of  the  proposed 
equation. 

Dividing  xH3x''+x=— 3x— 2  by  (x+l)^  or  ar+2x  +  l,  we  have 
a'+x— 2,  which  placed  equal  to  zero,  gives  the  two  roots  x^l, 
x=— 2,  or  the  two  factors  x— 1  and  x+2.     Hence  we  have 
x'  +  3x3+x2-3x-2=(x  +  l)=(x-l)  (x+2). 
Therefore  the  first  member  of  the  proposed  equation  is  equal  to 
{x-\-iy{x-l)%x+2f; 
or  the  proposed  equation  has   three  roots  equal  to— 1,  two  equal 
to  +1,  and  two  equal  to  —2. 
Take  the  examples, 
1st.  x^-7x''  +  10x5+22x^-43r''-35x'2  +  48x+36=0, 

(a;_2)2(a;-3)-(x  +  l)='=0. 
2d.  x^-3x«  +  9x'-19x''  +  27x^-33.i-  +  27,r-9  =  0, 

(x-l)%i-  +  3f=0. 
286.  When,  in  the  application  of  the  above  method,  we  obtain 


RESOLUTION  OF  .NU:.IERICAI.   EQUATIONS.  325 

an  equation  D=0,  of  a  degree  superior  to  the  second,  since  this 
equation  nnay  itself  be  subjected  to  the  method,  we  are  often  able 
to  decompose  D  into  its  factors,  and  in  this  way  to  find  the  different 
species  of  equal  roots  contained  in  the  equation  X=0,  and  the  num. 
ber  of  roots  of  each  species.  As  to  the  simple  roots  of  X  =  0,  we 
begin  by  freeing  this  equation  from  the  equal  factors  contained  in 
it,  and  the  resulting  equation,  X'=0,  will  make  known  the  simple 
roots. 


CHAPTER  VII. 


Resolution  of  Numerical  Equations,  involving  one  or 
more  Unknown  Quantities. 

297.  The  principles  established  in  the  preceding  chapter,  are  ap- 
plicable to  all  equations,  whether  their  co-efficients  are  numerical 
or  algebraic,  and  these  principles  should  be  regarded  as  the  ele- 
ments which  have  been  employed  in  the  resolution  of  equations  of 
the  higher  degrees. 

It  has  been  said  already,  that  analysts  have  hitherto  been  able  to 
resolve  only  the  general  equations  of  the  third  and  fourth  degree. 
Thp  formulas  they  have  obtained  for  the  values  of  the  unknown 
quantities  are  so  complicated  and  inconvenient,  when  they  can  be 
applied,  (which  is  not  always  possible),  that  the  problem  of  the  re- 
solution  of  algebraic  equations,  of  any  degree  whatever,  may  be 
regarded  as  more  curious  than  useful.  Therefore,  analysts  have 
principally  directed  their  researches  to  the  resolution  of  numerical 
equations,  that  is,  to  those  which  arise  from  the  algebraic  translation 
of  a  problem  in  which  the  given  quantities  are  particular  numbers ; 
and  methods  have  been  found,  by  means  of  which  we  can  always 
determine  the  roots  of  a  numerical  equation  of  any  given  degree. 

It  is  proposed  to  develop  these  methods  in  this  chapter. 
28 


326  ALGEBRA. 

To  render  the  reasoning  general,  we  will  represent  the  proposed 
equation  by 

m  which  P,  Q  . . .  denote  particular  numbers,  real,  positive,  or  ne- 
gative. 

First  Principle. 

288.  When  two  numbers  p  and  q,  substituted  in  the  place  of  x  in 
a  numerical  equation,  give  two  results,  affected  with  contrary  signs, 
the  proposed  equation  contains  a  real  root,  comprehended  between  these 
two  numbers 

Let  the  proposed  equation  be 

a;"'+Px'"-'+Qx'"-2+  ^  _  ^  Ta;+U=0. 

The  first  member  will,  in  general,  contain  both  positive  and  ne- 
gative terms ;  denote  the  sum  of  the  positive  terms  by  A,  and  the 
sum  of  the  negative  terms  by  B,  the  equation  will  then  take  the 
form 

A-B=0. 

Suppose  p<^q,  and  that  p  substituted  for  x  gives  a  negative  result, 
and  q  a  positive  result. 

Since  the  first  member  becomes  negative  by  the  substitution  of;?, 
and  positive  by  the  substitution  of  q,  it  follows  that  we  have  in  the 
first  case  A<B,  and  in  the  second  A>B.  Now  it  results  from  the 
nature  of  the  quantities  A  and  B,  that  they  both  increase  as  x  in- 
creases, since  they  contain  only  positive  numbers,  and  positive  and 
entire  powers  of  x ;  therefore,  by  making  x  augment  by  insensible 
degrees,  from  p  to  q,  the  quantities  A  and  B  will  also  increase  by  in- 
sensible  degrees.  Now  smce  A,  by  hypothesis,  from  being  less  than 
B,  afterwards  becomes  greater  than  it,  A  must  necessarily  have  a 
more  rapid  increment  than  B,  which  insensibly  destroys  the  excess 
that  li  liad  over  A,  and  finally  produces  an  excess  of  A  over  B. 
From  this,  we  conceive  that  in  the  passage  from  A<B  to  A>B, 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  327 

there  must  be  an  intermediate  value  for  which  A  becomes  equal  to 
B,  and  the  value  which  produces  this  result  is  a  root  of  the  equa- 
tion,  since  it  verifies  A— B=0,  or  the  proposed  equation.  Hence, 
the  proposition  is  proved. 

In  the  preceding  demonstration,  p  and  q  have  been  supposed  to 
be  positive  numbers ;  but  the  proposition  is  not  less  true,  whatever 
may  be  the  signs  with  which  p  and  q  are  affected.  For  we  will  re- 
mark,  in  the  first  place,  that  the  above  reasoning  applies  equally  to 
the  case  in  which  one  of  the  numbers  p  and  q,  p  for  example,  is  0 ; 
that  is,  it  could  be  proved,  in  this  case,  that  there  was  at  least  one 
real  root  between  0  and  q. 

Let  both  p  and  q  be  negative,  and  represent  them  by  — p'  and 
-q'. 

If,  in  the  equation 

x^'+Pa;'"-'+Qa;'"-»+  .  .  .  Ta;+U=0, 
we  change  x  into  —y,  which  gives  the  transformation 

(_3/)".+p(_y)-»+Q(_y)-2+  .  .  .  T(-t/)+U  =  0, 

it  is  evident  that  substituting  —p'  and  —q'  in  the  proposed  equation, 
amounts  to  the  same  thing  as  substituting  p'  and  q'  in  the  transfor- 
mation, for  the  results  of  these  substitutions  are  in  both  cases 

(-p')'"+P(-p')'"-^+Q(-p')"-^+  .  . .  T(-p')+U, 
and  (_5')'"+P(-j'r-^+Q(-?'r  ■'+  •  •  •  T(-?')+U  ; 
Now,  since  |>  and  q,  or  —p'  and  —q',  substituted  in  the  proposed  equa- 
tion, give  results  with  contrary  signs,  it  follows  that  the  numbers  p' 
and  q,  substituted  in  the  transformation,  also  give  results  with  con- 
trary signs ;  therefore,  by  the  first  part  of  the  proposition,  there  is 
at  least  one  real  root  of  the  transformation  contained  between  p' 
and  q' ;  and  in  consequence  of  the  relation  x=  —y,  there  is  at  least 
one  value  of  x  comprehended  between  —p'  and  —q',  or  p  and  q. 
This  demonstration  applies  to  cases  in  which  ^=0  or  ^=0. 

Lastly,  suppose  jj  po«<ive  and  q  negative  or  equal  to  —q':  by 
making  x=0  in  the  equation^  the  first  member  will  reduce  to  its 


328  ALGEBRA. 

last  term,  which  is  necessarily  affected  with  a  sign  contrary  to  that 
of  p,  or  that  of  —q';  whence  we  may  conclude  that  there  is  a  root 
comprehended  between  0  and  p,  or  between  0  and  —q',  and  conse- 
quently between;?  and  —q. 

Second  Principle. 

289.  When  two  numbers,  substituted  in  place  of  x,  in  an  equa- 
tion, give  results  affected  with  contrary  signs,  we  may  conclude  that 
there  is  at  least  one  real  root  comprehended  between  them,  but  we 
are  not  certain  that  there  are  no  more,  and  there  may  be  any  odd 
number  of  roots  comprised  between  them.  We  therefore  enunciate 
the  second  principle  thus. 

When  an  uneven  numler  (27t  +  l)  of  the  real  roots  of  an  equation, 
are  comprehended  hetwccn  tioo  numbers,  the  results  obtained  by  sub- 
stituting  these  numbers  for  x,  are  affected  with  contrary  signs,  and  if 
they  comprehend,  an  even  number  2n,  the  results  obtained  by  their  sub- 
stitution are  necessarily  affected  ivilh  the  same  sign. 

To  make  this  proposition  as  clear  as  possible,  denote  those  roots 
of  the  proposed  equation,  Xr=0,  which  are  supposed  to  be  compre- 
hended between  p  and  q,  by  a,  b,  c,  .  .  .,  and  by  Y,  the  product 
of  the  factors  of  the  first  degree,  with  reference  to^x,  correspond- 
ing both  to  those  real  roots  which  are  not  comprised  between  tlieni 
and  to  the  imaginary  roots ;  the  signs  of  p  and  q  being  arbitrary. 

The  first  member,  X,  can  be  put  under  the  form 
^x-a){x-b){x-c).  ..  xY. 

Now  substitute  in  X,  or  the  preceding  product,  p  and  q  in  place  of 
X ;   we  shall  obtain  tlie  two  results 

^p.a)(p-b)(p-c)...  xY', 

^q-a){q-b){q-c).  .  .   xY", 

Y'  and  Y"  representing  what  Y  becomes,  when  we  replace  x  by  p 
and  q ;  these  two  quantities  are  necessarily  affected  with  the  same 
sign,  for  if  they  were  not,  by  the  first  principle  Y=0  would  give  at 


KESOLUTION  OF  NUMERICAL  EQUATIONS.  329 

least  one  real  root  comprised  between  p  and  q,  which  is  contrary  to 
the  hypothesis. 

To  determine  the  signs  of  the  above  results  more  easily,  divide 
the  first  by  the  second,  we  obtain 

(p-«)  (P-^)  (P-c)  .  .  .  xY' 


(?- 

-a)  {q  - 

-b){q~c) 

.  .  .  XY" 

which  can 

be  written  thus ; 

1- 

-a     q- 

■  b     p—c 
■0     q—c 

Y' 

•••    Y" 

Now,  since  the  roots  a,  b, 

c,  .  .  .  are 

comprised  t 

we  have 

.>. 

,  b,  c,  d  .  . 

•> 

but 

< 

,  b,  e,  d.. 

•  5 

whence  we  deduce 

P- 

-a,  p- 

■b,  p-c,  .  . 

.>o, 

and 

1- 

-a,  q- 

-b,  q  —  c,  .  . 

■> 

hence,  since  p—a  and  q  —  a  are  affected  with  contrary  signs,  as  well 
Q.sp—b  and  q—b,p  —  c  and  q—c  .  .  .,  the  partial  quotients 

p-a     p-b      p-c    ^ 

q  —  a''     q  —  b'     q—c' 

Y' 

are  all  negative ;  moreover  -  is  essentially  positive,  since   Y' 

and  Y"  are  affected  with  the  same  sign  ;  therefore  the  product 

p—a     p—b     p  —  c  Y' 

X^—tX  X  •  .  .  -^r^, 

q—a     q—b      q—c  l" 

will  be  negative,  when  the  number  of  roots,  a,  b,  c  .  .  .,  compre- 
bended  between  p  and  q,  is  uneven,  and  positive  when  the  number  is 
even. 

28* 


330  ALG£BRA. 

Consequently,  the  two  results  (p—a)  (p—b)  (p— c)  .  .  .  xY', 
and  (q—a)  (q—b)(<l—c)  .  .  .  xY",  will  have  contrary  or  the  same 
signs,  according  as  the  number  of  roots  comprised  between  p  and  q 
is  uneven  or  even. 

Limits  of  the  real  Roots  of  Equations. 

290.  The  different  methods  for  resolving  numerical  equations, 
consist  generally  in  substituting  particular  numbers  in  the  proposed 
equation,  in  order  to  discover  if  these  numbers  verify  it,  or  whether 
there  are  roots  comprised  between  these  numbers.  But  by  reflect- 
mg  a  little  upon  the  composition  of  the  first  member,  the  first  term 
being  positive,  and  affected  with  the  highest  power  of  x,  which  is 
greater  with  respect  to  that  of  the  inferior  degree  in  proportion  to 
the  value  of  x,  we  are  sensible  that  there  are  certain  numbers, 
above  which  it  would  be  useless  to  substitute,  because  all  of  these 
numbers  would  give  positive  results. 

291.  Every  number  which  exceeds  the  greatest  of  the  positive 
roots  of  an  equation,  is  called  a  superior  limit  of  the  positive  roots. 

From  this  definition,  it  follows  that  the  limit  is  susceptible  of  an 
infinite  number  of  values;  for  when  a  number  is  found  to  exceed 
the  greatest  positive  root,  every  number  greater  than  this,  is,  for  a 
still  stronger  reason,  a  superior  limit.  But  it  may  be  proposed  to 
determine  the  simplest  possible  limit.  Now  we  are  sure  of  having 
one  of  the  limits,  when  we  obtain  a  number,  ivhich,  substituted  i7i 
fldce  of  X  renders  the  first  metnber  positive,  and  which,  at  the  same 
time,  is  such,  that  every  greater  nuviber  will  also  give  a  positive 
result. 

We  will  determine  such  a  number. 

292.  Before  resolving  this  question,  we  will  propose  a  more  sim- 
pie  one.  viz. 

To  determine  a  number,  which,  substituted  in  place  of  x  in  an 
equation,  will  render  the  first  term  x"  greater  than  the  arithmetical 
sum  of  all  the  others. 


LI3IITS  OF  ROOTS  OF  EQUATIONS.  331 

Suppose  that  all  the  terms  of  the  equation  are  negative,except  the 
first,  so  that 

af^^^jf-i-Qir-'-  .  .  .  _Tx-U=0. 

It  is  required  to  find  a  aiumber  for  x  which  will  render 

a;"'>Px'"-»+Q'"-='+  .  .  .  +Ta;+U. 

Let  k  denote  the  greatest  co-efficient,  and  substitute  it  in  place  of 
the  co-efficients  ;  the  inequality  will  become 

It  is  evident  that  every  number  substituted  for  x  which  will  satisfy 
this  condition,  will  for  a  stronger  reason,  satisfy  the  preceding.  Now, 
dividing  this  inequality  by  x"",  it  becomes 

k       k       k  k         k 

k 
Makmg  x=k,  the  second  member  becomes—,  or  1  plus  a  series 

of  positive  fractions ;  then  the  number  k  will  not  satisfy  the  ine- 
quality ;  but  by  supposing  x=^k+\,  we  obtain  for  the  second  mem- 
ber  the  series  effractions 

k  k  k  k  k 


■i+l+(A:+l)2+  {k+lf   +  •  •  •  +(i-4-ir-^+  {k+ir  ' 

which,  considered  in  an  inverse  order,  is  an  increasing  geometrical 

]^ 
progression,  the  first  term  of  which  is       .  t\„.  >  the  ratio  ^+1,  and 

k 
the  last  term  ,     ,  ;  hence  the  expression  for  the  sum  of  all  the 

terms  is,  (Art.  223), 


^-^■'^""-Wir  „,, 


k+i-i  {k+iy 

rhich  Ls  evidently  less  than  unity. 


332  ALOCBRA. 

Any  number  ^k-\~l,  put  in  place  of  x,  will  render  the  sum  of  the 

k       k 

fractions f— t+  .  •  •  still  less.     Therefore, 

X       ar 

The  greatest  co-efficient  of  the  equation  plus  unity,  or  any  greater 

number,  being  substituted  for  x,  will  render  the  first  term  x"*  greater 

than  the  arithmetical  sum  of  all  the  others. 

Ordinary  limit  of  the  Positive  Roots. 

293.  The  number  obtained  above  may  be  considered  a  prime 
limit,  since  this  number,  or  any  greater  number,  rendering  the  first 
term  superior  to  the  sum  of  all  the  others,  the  results  of  the  sub- 
stitution of  these  numbers  for  x  must  be  constantly  positive  ;  but 
this  limit  is  commonly  much  too  great,  because,  in  general,  the 
equation  contains  several  positive  terms.  We  will,  therefore,  seek 
for  a  limit  suitable  for  all  equations. 

Let  a;"*""  denote  the  power  of  x,  corresponding  to  the  first  nega- 
tive term  which  follows  x",  and  we  will  consider  the  most  uniavour- 
able  case,  viz.  that  in  which  all  of  the  succeeding  terms  are  nega- 
tive, and  affected  with  the  greatest  of  the  negative  co-efficients  in 
the  equation. 

Let  S  be  this  co-efficient,  and  try  to  satisfy  the  condition 
a;">Sa-"'-''+Sx'"-"-i+  .  .  .  Sx+S  ; 
or,  dividing  both  members  of  this  inequality  by  nf, 

S_       S        _S_  _S_     S_ 

Now  by  supposing  x''=S  or  x=  "V^,  the  second  member  be- 

S 
comes  -^,  or  1,  plus  a  series  of  positive  fractions  ;  but  by  making 

x=  VS  +  1,  or  (supposing,  for  simplicity,  VS  =  S',  whence  8  =  8'"), 
a:=S'  +  l,  the  second  member  becomes 


-+...  ,  ..„.,+ 


(S'  +  l)-.    '  (S'  +  l)"-^*^  •  •  •     '    (S'  +  ir-^        (S'  +  l)' 


#-*     U>    *y>SIJ[NARY  LIMIT  OF  THE  POSITIVE   ROOTS.  333 

.  .  y  s" 

which  is  a  ^ogression  by  quotients,  —^, — -—  being  the  first  term, 

S'" 
S'  +  l  the  ratio,  and  -7^7 — r-—  the  last  term.     Hence  the  expres- 
(b  +1)" 

sion  for  the  sum  of  all  these  fractions  is 

S'-  ^  S'" 

(S'  +  l)"*^^'  +  ^^~  (S'  +  l)"*  S'--*  S'"-» 


S'  +  l-l  (S'  +  l)"-'        (S'  +  lf 

which  is  evidently  less  than  1. 

Moreover,  every  number  >S'  +  1  or  \^S-\-l,  will,  when  substi- 

S         S 
tuted  for  x,  render  the  sum  of  the  fractions 1 Tr4-  •  •  •  •  still 

smaller,  since  the  numerators  remaining  the  same,  the  denominator 
will  increase.  Hence  VS  +  1,  and  any  greater  number,  will  ren- 
der  the  first  term  x"  greater  than  the  arithmetical  sum  of  all  the 
negative  terms  of  the  equation,  and  will  consequently  give  a  posi- 
tive result  for  the  first  member. 

Therefore  Vs  +  l,  or  wiity  increased  by  that  root  of  the  greatest 
negative  co-ejicient  icJiose  index  is  the  number  of  terms  which  precede 
the  first  negative  term,  is  a  superior  limit  of  the  positive  roots  of  the 
equation. 

Make  7i=l,  in  which  case  the  first  negative  term  is  the  second 
term  of  the  equation  ;  the  limit  becomes  VS  +  1,  or  S  +  1  ;  that  is, 
the  greatest  negative  co-efficient  plus  unity. 

Let  71  =  2,  then  the  two  first  terms  are  positive,  or  the  term  x"~' 
is  wanting  in  the  equation  ;   the  limit  is  then    V^S  +  l. 

When  71  =  3  the  limit  is  ^VS  +  l  .  .  . 

Find  the  superior  limits  for  the  positive  roots  in  the  following  ex- 
amples  : 
c4—5x'  +  37.r'- 3^+39=0;  VS  +  l='V~S'-\-l  =  6  ; 

a^+Tx*- 12x2- 49ar'+52x- 13=0  .      Vs4.1=  V49+l  =  8  ; 


334  ALGEBRA. 


Vs+i=)^- 


ft?^^ 


a;^  +  llx=_25x-67=0;  VS  +  1  =  ^;)^  +  1  or  0 

„    A—  11 

3x3— 2ar'-lla:  +  4=0;  VS  +  1=—      +1  or  5. 

o 

Newton's  method  for  determining  the  smallest  limit  in  entire 
numbers. 

294.  Let  X=0,  be  the  proposed  equation  ;  if  in  this  equation  we 
make  x=x'+u,  x'  being  indeterminate,  we  shall  obtain  (Art.  274), 

X'+Y'u+yu->  .  .  .  +w"=0.      (1) 

Conceive,  that  after  successive  trials  we  have  determined  a  number 

Z' 

for  X,  which,  substituted  in  X',  Y',  —  .  .  .,  renders  all  these  co-effi- 
cients positive  at  the  same  time  ;  this  number  will  be  greater  than 
the  greatest  positive  root  of  the  equation  X=0. 

For,  the  co-efficients  of  the  equation  (1)  being  all  positive,  no 
positive  number  can  verify  it ;  therefore  all  of  the  real  values  of  u 
must  be  negative;  but  from  the  equation  a'=x'+if,  we  have  xi^=x—x' ; 
and  in  order  that  the  values  of  it  corresponding  to  each  of  the  values 
of  X  and  x'  (already  determined)  may  be  negative,  it  is  absolutely 
necessary  that  the  greatest  positive  value  of  x  should  be  less  than 
the  value  of  x'. 

EXAMPLE. 

x^-Sx'-ex^— 19x4-7=0. 
As  x'  is  indeterminate,  the  letter  x  may  be  retained  in  the  forma- 
tion of  the  derived  polynomials,  and  we  have 

X    ^x^-Sr'— 6x2-19x-|r7, 
Y    =:4r'-15r'-12x-19, 

Z 

_  =6x2-15x-6, 

V 

=4x— 6. 

2.3 


SMALLEST  LIMIT  OP  THE  ROOTS.  335 

The  question  is,  as  stated  above,  reduced  to  finding  the  smallest 
number  which,  substituted  in  place  of  a>,  will  render  all  of  these  po- 
lynomials positive^ 

It  is  plain  that  2  and  every  number  >2,  will  render  the  polyno- 
mial of  the  first  degree  positive. 

But  2,  substituted  in  the  polynomial  of  the  second  degree,  gives  a 
negative  result;  and  3,  or  any  number  >3,  gives  a  positive  result. 

Now  3  and  4,  substituted  in  the  polynomial  of  the  third  degree, 
give  a  negative  result ;  but  5  and  any  greater  number,  give  a  posi- 
tive result. 

Lastly,  5  substituted  in  X,  gives  a  negative  result,  and  so  does  6  ; 
for  the  three  first  terms  x*  — 5a;^— 6x^  are  equivalent  to  the  expres- 
sion 3^(x—5)  —  6aP,  which  is  reduced  to  0  when  x=6  ;  but  x=7  evi- 
dently gives  a  positive  result.  Hence  7  is  a  superior  limit  of  the 
positive  roots  of  tJie  proposed  equation ;  and  since  it  has  been  shown 
that  6  gives  a  negative  result,  it  follows  that  there  is  at  least  one 
real  root  between  6  and  7. 

Applying  this  method  to  the  equation 

a;5_3x''-8x''— 25x2  +  43;- 39=0, 

the  superior  limit  will  be  found  to  be  6. 

We  should  find  7,  for  the  superior  limit  of  the  positive  roots  of 
the  equation 

xs_5x*- 1.3x^  +  17x2-69=0. 

This  method  is  scarcely  ever  used,  except  in  finding  incommen- 
surable  roots. 

295.  It  remains  to  determine  the  superior  limit  of  the  negative 
roots,  and  the  inferior  limits  of  the  positive  and  negative  roots. 

Hereafter  we  shall  designate  the  superior  limit  of  the  positive  roots 
of  an  equation  by  the  letter  L. 

1st.  If  in  the  equation  X=0,  we  make  x=— y,  which  gives  the 
transformed  equation  Y=0,  it  is  clear  that  the  positive  roots  of  this 
new  equation,  taken  with  the  sign  — ,  will  give  the  negative  roots  of 


336  ALGEBRA. 

the  proposed  equation ;  therefore,  determining,  by  the  known  me- 
thods, the  superior  limit  L'  of  the  positive  roots  of  the  equation  Y=0, 
we  shall  have  —  L'  for  the  superior  limit  (numerically)  of  the  nega- 
tive roots  of  the  proposed  equation. 

2d.  If  in  the  equation  X  =  0,  we  make  j;= — ,   which    gives   the 

equation  Y=0,  it  follows  from  the  relation  x= —  that  the  greatest 

positive  values  of  y  correspond  to  the  smallest  of  x ;  hence,  desig- 
nating  the  superior  limit  of  the  positive  roots  of  the  equation  Y=0 

by  L",  we  shall  have     ^  „     for  the  inferior  limit  of  the  positive  rods 

of  the  proposed  equation, 

1 
3d.  Finally,  if  we  replace  x,  in  the  proposed  equation,  by , 

and  find  the  superior  limit  L"'  of  the  transformed  equation  Y  =  0, 

— :jr-;77-  will  be  the  inferior  limit  (numerically)  of  the  negative  roots 

of  the  proposed  equation. 


296.  Every  equation  in  which  there  are  no  variations  in  the . 
that  is,  in  which  all  the  terms  are  positive,  must  have  all  of  its  real 
roots  negative ;  for  every  positive  number  substituted  for  x  will  ren- 
der the  first  member  essentially  positive. 

Every  complete  equation,  having  its  terms  alternately  positive  and 
negative,  must  have  its  real  roots  all  positive  ;  for  every  negative 
number  substituted  for  x  in  the  proposed  equation,  would  render  all 
the  terms  positive,  if  the  equation  was  of  an  even  degree,  and  all  of 
them  negative  if  it  was  of  an  odd  degree.  Hence  the  sum  would 
not  be  equal  to  zero  in  either  case. 

This  is  also  true  for  every  incomplete  equation,  in  which  there 
results,  by  sulstituting  —  y  for  x,  an  equation  having  all  of  its  terms 
affected  with  the  same  sign. 


SMALLEST  LIMIT  OP  THE  ROOTS.  337 

Consequences  deduced  from  the  preceding  Principles. 

First. 

297.  Every  equation  of  an  odd  degree,  the  co-efficients  of  which 
are  real,  has  at  least  one  real  root  affected  with  a  sign  contrary  to 
thai  of  its  last  term. 

For,  let  a;'"+Pa;'"-'+  .  .  .  Ta;±U=0,  be  the  proposed  equation; 
and  first  consider  the  case  in  which  the  last  term  is  negative. 

By  making  .t=0  the  first  member  becomes  — U.  But  by  giving 
a  value  to  x  equal  to  the  greatest  negative  co-efficient  plus  unity,  or 
(K-f-1),  the  first  term  .r™  will  become  greater  than  the  arithmetical 
sum  of  all  the  others  (Art.  292),  the  result  of  this  substitution  will 
therefore  be  positive ;  hence,  there  is  at  least  one  real  root  compre- 
hended between  0  and  K  +  1,  which  root  is  positive,  and  consequently 
affected  with  a  sign  contrary  to  that  of  the  last  term. 

Suppose  now  that  the  last  term  is  positive. 

Making  .r=0,    we    obtain  +U    for  the  result;  but  by  putting 

—  (K  +  1)  in  place  of  a;,  we  shall  obtain  a  negative  result,  since  the 
first  term  becomes  negative  by  this  substitution ;  hence  the  equa- 
tion  has   at   least   one   real   root   comprehended   between  0  and 

—  (K+1),  which  is  negative,  or  affected  toith  a  sign  contrary  to  that 
of  the  last  term. 

Second. 

298.  Every  equation  of  an  even  degree,  involving  only  real  co. 
efficients  of  which  the  last  term  is  negative,  has  at  least  two  real  roots, 
one  positive  and  the  other  negative.  For,  let  —  U  be  the  last  term  ; 
making  x'=0,  there   results  — U.     Now  substitute  either  K  +  1,  or 

—  (K  +  1),  K  being  the  greatest  negative  co-efficient  of  the  equa- 
tion :  as  ?n  is  an  even  number,  the  first  term  x'^  will  remain  positive  ; 
besides,  by  these  substitutions,  it  becomes  greater  than  the  sum  of 
all  the  others ;  therefore  the  results  obtained  by  these  substitutions 
are  both  positive,  or  affected  with  a  sign  contrary  to  that  given  by 
the  hypothesis  x=0 ;  hence  the  equation  has  at  least  tioo  real  ro-^'v 

20 


33o  ALGEBKA. 

one  comprehended  between  0  and  K-j-l,  or  positive,  and  the  other 
between  0  and  —  (K  +  1),  or  negative. 

Third. 

299.  If  an  equation,  involving  only  real  co-efficients,  contains 
imaginary  roots,  the  number  of  these  roots  must  he  even. 

For,  conceive  that  the  first  member  has  been  divided  by  all  the 
simple  factors  corresponding  to  the  real  roots  ;  the  co-efficients  of  the 
quotient  will  be  real  (261) ;  and  the  equation  must  also  he  of  an  even 
degree;  for  if  it  was  uneven,  by  placing  it  equal  to  zero,  we  should 
obtain  an  equation  that  would  contain  at  least  one  real  root,  which, 
from  the  nature  of  the  equation,  it  cannot  have. 

Remark.  300.  There  is  a  property  of  the  above  polynomial  quo- 
tient which  belongs  exclusively  to  equations  containing  only  imagi- 
nary roots  ;  viz.  every  such  equation  ahvays  remains  positive  for  any 
real  value  substituted  for  x. 

For,  if  it  could  become  negative,  since  we  could  also  obtabi  a  posi- 
tive result,  by  substituting  K  +  1  or  the  greatest  negative  co-efficient 
plus  unity  for  x,  it  would  follow  that  this  polynomial  placed  equal 
to  zero,  would  have  at  least  one  real  root  comprehended  between 
K-fl  and  the  number  which  would  give  a  negative  result. 

It  also  follows,  that  the  last  term  of  this  polynomial  must  be  posi. 
live,  otherwise  a;=0  would  give  a  negative  result. 

Fout^th. 

301.  When  the  last  term  of  an  equation  is  positive,  the  number  of 
its  real  positive  roots  is  even  ;  and  when  it  is  negative  this  number  is 
uneven. 

For,  first  suppose  that  the  last  term  is  +U,  or  positive.  Since 
by  making  a;=0,  there  will  result  +11,  and  by  making  a;=K  +  l, 
the  result  will  also  be  positive,  it  follows  that  0  and  K  +  1  give  two 
results  affected  with  the  same  sign,  and  consequently  (289),  the 
numbor  of  real  roots,  (if  any),  comprehended  between  them,  is  even. 


Descartes'  rule.  339 

When  the  last  term  is  — U,  then  0  and  K+l  give  two  results 
affected  with  contrary  signs,  and  consequently  comprehend  either  a 
single  real  root,  or  an  odd  number  of  them. 

The  reciprocal  of  this  proposition  is  evident. 

Descartes^  Rule. 

302.  An  equation  of  any  degree  whatever  cannot  have  a  greater 
number  of  positive  roots  than  there  are  variations  in  the  signs  of  Us 
terms,  nor  a  greater  number  of  negative  roots  than  there  are  perma- 
nences of  these  signs. 

In  the  equation  x—a=0,  there  is  one  variation,  that  is  a  change 
of  sign  in  passing  along  the  terms,  and  one  positive  root,  x=a.  And 
in  the  equation  x+b=0,  there  is  one  permanence,  and  one  negative 
root,  x=—b. 

If  these  equations  be  multiplied  together,  there  will  result  an  equa- 
tion  of  the  second  degree, 

x^  —  a  I  x  —  ab 

If  fl  is  less  than  b,  the  equation  will  be  of  the  first  form  (Art.  144) ; 
and  if  a>J  the  equation  will  be  of  the  second  form  :  that  is 

a<i     gives     xr'-{-px—q=0     and 
a>i  x^—px—q  =  0 

In  either  case,  there  is  one  variation,  and  one  permanence,  and 
since  in  either  form,  one  root  is  positive  and  one  negative,  it  follows 
that  there  are  as  many  positive  roots  as  there  are  variations,  and 
as  many  negative  roots  as  there  are  permanences. 

The  proposition  would  evidently  be  demonstrated  in  a  general 
manner,  if  it  were  shown  that  the  multiplication  of  the  first  member 
by  a  factor  x— fl,corresponding  to  a  positive  root,  would  introduce  at 
least  one  variation,  and  that  the  multiplication  by  a  factor  x-\-a, 
would  introduce  at  least  one  permanence. 


:0. 


340  ALGEBRA. 

Let  there  be  the  equation 

af'±Ax'^»±Bx'"-2±Ca;""-3±  .  .  .  ±Ta:±U=0, 

in  which  the  signs  succeed  each  other  in  any  manner  whatever  ;  by 
niiiltiplying  it  by  x—a,  we  have 

-a   I      ipAa  I  zpBa  |  z^Ta  \    q=Ua 

The  co-efficients  which  form  the  first  horizontal  line  of  this  pro- 
duct, are  those  of  the  proposed  equation,  taken  with  the  same  sign  ; 
and  the  co-efficients  of  the  second  line  are  formed  from  those  of  the 
first,  multiplied  by  a,  taken  with  contrary  signs,  and  advanced  one 
rank  towards  the  right. 

Now,  so  long  as  each  co-efficient  of  the  upper  line  is  greater  than 
the  corresponding  one  in  the  lower,  it  will  determine  the  sign  of  the 
total  co-efficient ;  hence,  in  this  case  there  will  be,  from  the  first 
term  to  that  preceding  the  last,  inclusively,  the  same  variations  and 
the  same  perm.anences  as  in  the  proposed  equation  ;  but  the  last 
term  zpUa  having  a  sign  contrary  to  that  which  immediately  pre- 
cedes it,  there  must  be  one  or  more  variations  than  in  the  proposed 
equation. 

When  a  co-efficient  in  the  lower  line  is  affected  with  a  sign  con- 
trary to  the  one  corresponding  to  it  in  the  upper,  and  is  also  greater 
than  this  last,  there  is  a  change  from  a  permanence  of  sign  to  a 
variation;  for  the  sign  of  the  term  in  which  this  happens,  being  the 
same  as  that  of  the  inferior  co-efficient,  must  be  contrary  to  that  of 
the  preceding  term,  which  has  been  supposed  to  be  the  same  as  that 
of  its  superior  co-efficicnt.  Hence,  each  time  we  descend  from  the 
upper  to  the  lower  line,  in  order  to  determine  the  sign,  there  is  a 
variation  which  is  not  found  in  the  proposed  equation ;  and  if,  after 
passing  into  the  lower  line,  we  continue  in  it  throughout,  we  shall  find 
for  the  remaining  terms  the  samevariatipns  and  the  same  perma- 
nences as  in  the  proposed  equation,  sil^ce  tKe  co-efficients  of  this  line 
are  all  affected  with  signs  contrary  to  those  of  the  primitive  co-effi- 
cients.    This  supposition  would  therefore  give  us  one  variation  for 


DESCARTES     UULE. 


341 


each  positive  root.  But  if  we  ascend  from  the  lower  to  the  upper 
line,  there  may  be  either  a  variation  or  a  permanence.  But  even 
by  supposing  that  this  passage  produces  permanences  in  all  cases, 
since  the  last  term  zpUa  forms  a  part  of  the  lower  line,  it  will  be 
necessary  to  go  once  more  from  the  upper  line  to  the  lower,  than 
from  the  lower  to  the  upper.  Hence  the  new  equation  must  have  at 
least  one  more  variation  than  the  proposed ;  and  it  will  be  the  same 
for  each  positive  root  introduced  into  it. 

It  may  be  demonstrated,  in  an  analogous  manner,  that  the  multi- 
plication by  a  factor  x-\-a,  corresponding  to  a  negative  root,  would 
introduce  one  permanence  more.  Hence,  in  any  equation  the  num- 
ber of  positive  roots  cannot  be  greater  than  the  number  of  varia- 
tions of  sign,  nor  the  number  of  negative  roots  greater  than  the 
number  of  permanences. 

303.  Consequence.  When  the  roots  of  an  equation  are  all  real, 
the  number  of  positive  roots  is  equal  to  the  number  of  variations,  and 
the  number  of  negative  roots  is  equal  to  the  number  of  permanences. 

For,  let  m  denote  the  degree  of  the  equation,  n  the  number  of 
variations  of  the  signs,  p  the  number  of  permanences  ;  we  shall  have 
m=n-[-p.  Moreover,  let  n'  denote  the  number  of  positive  roots, 
and  p'  the  number  of  negative  roots,  we  shall  have  m=n'+p'; 
whence 

n-{-p=.n'  -\-p 

or,  n—n'=p'—p. 

Now,  we  have  just  seen  that  n'  cannot  be  >n,  and  p'  cannot  be  >_p  ; 

therefore  we  must  have  n'=n,  and  p'=p. 

Remark.  304.  When  an  equation  wants  some  of  its  terms,  we 
can  often  discover  the  presence  of  imaginary  roots,  by  means  of  the 
above  rule. 

For  example,  take  the  equation 

oi?-\-px-{-q=0, 

p  and  q  being  essentially  positive  ;  introducing  the  term  which  is 

29* 


842  ALGEBRA. 

wanting,  by  affecting  it  with  the  co-efficient  ±0,  it  becomes 

By  considering  only  the  superior  signs  we  should  only  obtain  per- 
manences, whereas  the  inferior  sign  would  give  two  variations. 
This  proves  that  the  equation  has  some  imaginary  roots ;  for  if  they 
were  all  three  real,  it  would  be  necessary  by  virtue  of  the  superior 
sign,  that  they  should  be  all  negative,  and,  by  virtue  of  the  inferior 
sign,  that  two  of  them  should  be  positive  and  one  negative,  which 
are  contradictory  results. 

We  can  conclude  nothing  from  an  equation  of  the  form 
x^—px-\-q=zO ; 
for  introducing  the  term  ±0.3^,  it  becomes 

x^=tO  .  x^—px+q=0, 
which  contains  one  permanence  and  two  variations,  whether  we  take 
the  superior  or  inferior  sign.  Therefore  this  equation  may  have  its 
three  roots  real,  viz.  two  positive  and  one  negative  ;  or,  two  of  its 
roots  may  be  imaginary  and  one  negative,  since  its  last  term  is  ne- 
gative (Art.  301). 

Of  the  Commensurable  Roots  of  Numerical 
Equations. 

305.  Every  equation  in  which  the  co-efficients  are  whole  num- 
bers, that  of  the  first  term  being  unity,  can  only  have  whole  num- 
bers for  its  commensurable  roots. 

For,  let  there  be  the  equation 

a;-+Pa;"-i+Qa;"— ^+  .  .  .  +T.r  +  U  =  0  ; 

in  which  P,  Q  .  .  .  T,  U,  are  whole  numbers,  and  suppose  that  it 

a 
could  have  a  commensurable  fraction  -j-  for  a  root.     Substituting 

this  fraction  for  x,  the  equation  becomes 


COMMENSURABLE  ROOTS  OP  EQUATIONS.  343 

whence,  multiplying  the  whole  equation  by  Z*""',  and  transposing, 

^=_Pa--i_Qa— ^'5-  .  .  .  -Tai'^-'-Vb'^-'  ; 
o 

but  the  second  member  of  this  equation  is  composed  of  a  series  of 
entire  numbers,  whilst  the  first  is  essentially  fractional,  for  a  and  b 
being  prime  with  each  other,  a"  and  b  will  also  be  prime  with  each 
other  (Art.  118),  hence  this  equality  cannot  exist ;  for,  an  irreduci- 
ble  fraction  cannot  be  equal  to  a  whole  number. 

Therefore  it  is  impossible  for  any  commensurable  fraction  to  sa- 
tisfy  the  equation.  Now  it  has  been  shown  (Art.  277),  that  an 
equation  containing  rational,  but  fractional  co-efficients,  can  be 
transformed  into  another  in  which  the  co-efficients  are  whole  num- 
bers, that  of  the  first  term  being  unity.  Hence  the  research  of  the 
commensurable  roots,  entire  or  fractional,  can  alicays  be  reduced  to 
that  of  the  entire  roots. 

306.  This  being  the  case,  take  the  general  equation 

^m_^Pa;m-i+Qa;— 2+  .  .  .  +R,^''+Sr'+Ta;+U=0, 

and  let  a  denote  any  entire  number,  positive  or  negative,  which  will 
verify  it. 

Since  a  is  a  root,  we  shall  have  the  equation 

a'"+Pa"-»+  .  .  .  +Ra='+Sa^+Ta+U=0  .  .  •  (1) ; 

replacing  a  by  all  the  entire  positive  and  negative  numbers  between 
1  and  the  limit  +L,  and  between  —1  and  — L',  those  which  verify 
the  above  equality  will  be  the  roots  of  the  equation.  But  these 
trials  being  long  and  troublesome,  we  will  deduce  from  equation  (1), 
other  conditions  equivalent  to  this,  and  easier  verified. 

Transposing  all  the  terms  except  the  last,  and  dividing  by  a,  the 
equation  (1)  becomes 

— =-a"-^-Pa"-2_  .  .  .  _Ra2_Srt-T  .  .  .  (2) ; 
a 

now,  the  second  member  of  this  equation  is  an  entire  number,  hence 


344  ALGEBRA. 

—  must  be  an  entire  number ;  therefore  the  entire  roots  of  the  equa- 
tion are  comprised  among  the  divisors  of  the  last  term. 

Transposing  —  T  in  the  equation  (2)  and  dividing  by  a,  and  ma- 

U 
king |-T=T';  it  becomes 

T' 

—  =-a"'-2-Pa"'-3  .  .  .  -Ra-S  .  .  .  (3) ; 

T' 

the  second  member  of  this  equation   being  an  entire  number,  — 

U' 

or,  the  quotient  of  the  division  of \-T  by  a,  is  an  entire  numler. 

Transposing  the  term  —  S  and  dividing  by  a,  it  becomes,  by  sup- 

T' 
posing f-S=S', 

?-=_a-3_Pam-4_  ,  .  .  _R  .  .  .  (4), 
a 

S' 
the  second  member  of  this  equation  being  an  entire  number,  — 

T' 

or,  the  quotient  of  the  division  of \-^  by  a,  is  an  entire  number. 

By  continuing  to  transpose  the  terms  of  the  second  member  into 

the  first,  we  shall,  after  m— 1  transformations,  obtain  an  equation  of 

Q' 

the  form — =— a— P, 

a 

Then,  transposing  the  term    —  P,   dividing  by  a,  and  making 

Q'  F  P' 

[-P=P'»  we  shall  find  —=-1,  or [-1=0. 

a  a  a 

This  equation,  which  is  only  a  transformation  of  the  equation  (1), 
is  the  last  condition  which  it  is  requisite  and  necessary  that  the  en- 
tire  number  a  should  satisfy,  in  order  that  it  may  be  known  to  be  a 
root. 

From  the  preceding  conditions  we  may  conclude  that,  in  order 


COMMENSURABLE  ROOTS  OP  EQUATIONS.  345 

that  an  entire  number  a,  positive  or  negative,  may  be  a  root  of  the 
proposed  equation,  it  is  necessary 

That  the  quotient  of  the  last  term,  divided  by  a,  should  be  an  en- 
tire number ; 

Adding  to  this  quotient  the  co-efficient  of  .r',  taken  with  its  sign, 
the  quotient  of  this  sum  divided  by  a,  must  be  entire  ; 

Adding  the  co-efficient  of  x^  to  this  quotient,  the  quotient  of  this 
new  sum  by  a,  must  be  entire;  and  so  on. 

Finally,  adding  the  co-efficient  of  the  second  term,  or  of  a;"""',  to 
the  preceding  quotient,  the  quotient  of  this  sum  divided  by  a,  7nust  be 
entire  and  eqiud  to  —1  •  or,  the  result  of  the  addition  of  unity,  or  the 
co-efficient  of  x"\  to  the  preceding  quotient,  must  be  equal  to  0. 

Every  number  which  will  satisfy  these  conditions  will  be  a  root, 
and  those  which  do  not  satisfy  them  should  be  rejected. 

All  the  entire  roots  may  be  determined  at  the  same  time,  as  fol- 
lows. 

After  having  determined  all  the  divisors  of  the  last  term,  tcrite 
those  which  are  comprehended  between  the  limits  +L  and  —  L'  upon 
the  same  horizontal  line ;  then  underneath  these  divisors  write  the  quo- 
tients  of  the  last  term  by  each  of  them. 

Add  the  co-efficient  of  x'  to  each  of  these  quotients,  and  write  the 
sums  underneath  the  quotients  ichich  correspond  to  them  ;  then  divide 
these  siuns  by  each  of  the  divisors,  and  write  the  quotients  underneath 
the  corresponding  sums ;  taking  care  to  reject  the  fractional  quo- 
tients and  the  divisors  which  produce  them  ;  and  so  on. 

When  there  are  terms  wanting  in  the  proposed  equation,  their 
co-efficients,  which  are  to  be  regarded  as  equal  to  0,  must  be  taken 
into  consideration. 


x*—a.'^—13x=  + 16a:— 48=0. 
The  superior  limit  of  the  positive  roots  of  this  equation  is  13-f  1 
or  14  (Art.  293).     The  co-efficient  48  is  not  considered,  since  the 


346  ALGEBRA. 

two  last  terms  can  be  put  under  the  form  16(x— 3) ;  hence  when 
a;>3  this  part  is  essentially  positive. 

The  superior  limit  of  the  negative  roots  is  —(1+  V48),  or  —8 
(Art.  295). 

Therefore,  the  divisors  are  1,  2,  3,  4,  6,  8,  12  ;  moreover,  neither 
-f  1,  nor  —1,  will  satisfy  the  equation,  because  the  co-efficient  —48 
is  itself  greater  than  the  sum  of  all  the  others ;  we  should  therefore 
try  only  the  positive  divisors  from  2  to  12,  and  the  negative  divisors 
from  —2  to  —6  inclubi -ely. 

By  observing  the  rule  given  above,  we  have 


12, 

8, 

6, 

4, 

3, 

2, 

—   2, 

-   3, 

-   4, 

-   6 

-   4, 

-  6, 

-8, 

-12, 

-16, 

-24, 

+24, 

+  16, 

+  12, 

+  8 

+  12, 

+  10, 

+8, 

+   4, 

0, 

-   8, 

+  40, 

+  32, 

+  28, 

+  24 

+   1, 

.., 

+   1, 

0, 

-   4, 

-20, 

.., 

-   7, 

-   4 

-12, 

••J 

-12, 

-13, 

-17, 

-33, 

.., 

-20, 

—  17 

-    1, 

.., 

-   3, 

.., 

.., 

.., 

••> 

+  5, 

.. 

-   2, 

-  4, 

—  1, 

-, 

•  •> 

••' 

••' 

+  4, 
—   1, 

•• 

The  Jirst  line  contains  the  divisors,  the  second  contains  the  quo- 
tients of  the  division  of  the  last  term  —48,  by  each  of  the  divisors. 
The  third  line  contains  these  quotients  augmented  by  the  co-efficient 
+  16,  and  the  fourth  the  quotients  of  these  sums  by  each  of  the  di- 
visors  ;  this  second  condition  excludes  the  divisors  +8,  +6,  and  —3. 

The  fifth  is  the  preceding  line  of  quotients,  augmented  by  the  co- 
efficient —  13,  and  the  sixth  is  the  quotients  of  these  sums  by  each 
of  the  divisors ;  this  third  condition  excludes  the  divisors  3,  2,  —2 
and  —6. 

Finally,  the  seventh  is  the  third  line  of  quotients,  augmented  by 
the  co-efficient  —1,  and  the  eighth  is  the  quotients  of  these  sums  by 
each  of  the  divisors.  The  divisors  +4  and  — 4  are  the  only  ones 
which  give  —1  ;  hence  +4  and  —4  are  the  only  entire  roots  of 
the  equation. 

In  fact,  if  we   divide  a;*— r"- 13ar'+16x— 48,  by  the  product 


INCOMMENSURABLE  ROOTS.  347 

(«— 4)  (x+4),  or  x'—ie,  the  quotient  will  be  x^—x+S,  which 
placed  equal  to  zero,  gives 


therefore,  the  four  roots  are 

EXAMPLES. 

1st.  X'*  — 5a;='+25a;— 21  =  0. 

2d.  15a:^-19a;''+6af'+15ic2— 19a;+6=0. 

3d.  9x<'+30x5+22x*+10af'+17ar'— 20a;+4=0. 

Of  Real  and  Incommensurable  Roots. 

307.  When  an  equation  has  been  freed  from  all  the  divisors  of 
the  first  degree  which  correspond  to  its  commensurable  roots,  the 
resulting  equation  contains  the  incommensurable  roots  of  the  pro- 
posed equation,  either  real  or  imaginary. 

The  true  form  of  the  real  incommensurable  roots  of  an  equation 
will  remain  unknown,  so  long  as  there  is  not  a  general  method  for 
resolving  equations  of  the  higher  degrees.  AUhough  this  problem 
has  not  been  resolved,  yet  there  are  methods  for  approximating  as 
near  as  we  please  to  the  numerical  values  of  these  roots. 

We  shall  here  consider  only  the  case  in  which  tlie  difference  be- 
tween any  two  roots  of  the  proposed  equation  is  greater  than  unity, 
omitting  as  too  difficult  for  an  elementary  treatise,  the  cases  in  which 
this  difference  is  less  than  unity. 

We  will  also  suppose,  in  what  follows,  that  we  have  obtained  the 
narrowest  limits  4-L  and  — L',  by  Newton's  method  (Art.  294). 

308.  Each  of  the  incommensurable  roots  being  necessarily  com- 
posed  of  an  entire  part  and  a  part  less  than  unity,  we  shall  first  deter- 
mme  the  entire  part  of  each  root. 


348  ALGEBRA. 

For  this  purpose,  it  is  necessary  to  substitute,  in  the  equation,  for 
X,  the  series  of  natural  numbers  0, 1,  2,  3  . . .  and  —  1,  —  2,  —  3  . . ., 
comprised  between  4-L  and  — L'.  Since  there  must  be  a  real  root 
between  two  numbers,  which,  by  their  substitution  produce  results 
affected  with  different  signs  (Art.  288),  it  follows  that  each  pair  of 
consecutive  numbers  giving  results  affected  with  contrary  signs,  toill 
comprehend  a  real  root,  and  but  one,  since  by  hypothesis  the  difference 
between  any  two  of  the  roots  is  greater  than  unity.  The  entire 
part  of  the  root  will  be  the  smallest  of  the  two  numbers  substituted. 

There  are  two  cases  which  may  occur;  viz.  by  these  diffei'ent 
substitutions  there  may  be  as  many  changes  of  sign  as  there  are 
units  in  the  degree  of  the  equation ;  in  which  case  we  may  con- 
clude that  all  the  roots  are  real.  Or,  the  number  of  changes  of  the 
sign  will  be  less  than  the  degree  of  the  equation,  and,  in  this  case, 
it  will  have  as  many  real  roots  as  there  are  changes  of  sign ;  the 
other  roots  will  be  imaginary.  In  both  cases,  this  method  makes 
known  the  entire  part  of  each  of  the  real  roots. 

It  now  remains  to  determine  the  part  which  is  less  than  unity. 

Newton's  Method  of  Approximation. 

809.  In  order  that  this  method  may  be  more  easily  comprehend- 
ed, we  shall  take  the  equation 

a;3_5a;-3=0  .  .  .  (1). 

The  superior  limits  of  the  positive  and  negative  roots  being  -f-3 
and  —2,  we  make 

x=  —  2,  —1,  0,  1,  2,  3; 
whence  a;=  — 2  the  result  is  —1, 

x=-\       ...        +1, 


x=     0      .     . 

.        -3, 

a:=      1       .      . 

—  "7, 

x=     1      .     . 

.        -5, 

x=     3       .     . 

.        +9. 

Newton's  method  of  approximation.  349 

As  tliere  are  three  changes  of  sign,  it  follows  that  the  three  roots 
of  the  equation  are  real ;  viz.  ojie  positive  contained  between  2  and 
3,  two  negative,  one  of  which  is  contained  between  0  and  —1,  the 
other  between  —1  and  —2. 

We  shall  first  consider  the  positive  value  between  2  and  3. 

The  required  root  being  between  2  and  3,  we  will  try  to  contract 
these  limits,  by  taking  the  mean  2i,  or  2,5,  and  substituting  it  in 
the  equation  a;^— 5x— 3=0  ;  the  result  of  which  is  +0,125.  Now 
2  has  already  given  —5  for  a  result,  therefore  the  root  is  between 
2  and  2,5. 

We  will  now  consider  another  number,  between  2  and  2,5 ;  but 
as,  from  the  results  given  from  2  and  2,5,  it  is  to  be  presumed  that 
the  root  is  nearer  2,5  than  2,  suppose  x=2,4 ;  we  shall  obtain 
—  1,176;  whereas  2,5  has  given  +0,125.  Therefore  the  root  is 
between  2,4  and  2,5. 

By  contmuing  to  take  the  means,  we  should  be  able  to  contract 
the  two  limits  of  the  roots  more  and  more.  But  when  we  have 
once  obtained,  as  in  the  above  case,  the  value  of  x  to  at  least  0,1, 
we  may  approximate  nearer  in  another  way,  and  it  is  in  this  that 
Newtoti's  method  principally  consists. 

In  the  equation  a;^  — 5a'— 3  =  0,  make  a;=2,4+M. 

There  will  result  (Art.  274),  the  transformation 

Z' 

X'+Y'it  +  — w2+u^=0; 

in  which  X' =(2,4)^— 5(2,4)-3=  — 1,176, 

Y' =3(2,4)2-5=12,28, 

Z' 

— -  =  3(2,4)  =  7,2. 

The  equation  involving  u,  being  of  the  third  degree,  cannot  be 
resolved  directly,  but  by  transposing  all  the  terms  except  Y'm,  and 
dividing  both  members  by  Y',  it  can  be  put  under  the  form 
X'       Z'  1 

"=-y^-2:y"'-y^"^- 

30 


350  ALGEBRA. 

This  being  the  case,  since  one  of  the  three  roots  of  this  equation 
must  be  less  than  — ,  from  the  relation  ic=2,4+M,  the  correspond- 
ing values  of  u^  and  v?  are  less  than  and  .     Moreover, 

the  inspection  of  the  numerical  values  of  Y'  and  Z',  proves  that 

Z' 

is  <1  ;  therefore  the  value  of  u  only  differing  numerically 

X'  Z'  1 

from   —^,  by  the  quantity  v?+~u^,  (which  most  frequently 

1    \  X'  .    . 

is  less  than  J,  is  expressed  by  ——to  withm  1,01. 

As,  in  this  example, 

X'       +1,176         1176 


12,28  12280 


:0,09 


there  will  result  «=0,09,  to  within  ,  and  consequently 

1 

a:=  2,4  4- 0,0  9= 2,49,     to  within     — — . 

In  fact,  2,49  substituted  in  the  first  member  of  the  proposed  equa- 
tion, gives  —0,011751  ; 
whilst  2,250  gives  +0,125. 

To  obtain  a  new  approximation,  make  a;=2,49+w'  in  the  pro- 
posed  equation,  and  we  have 

X"+Y'V+— «'H«'^=0; 

in  which         X"=  (2,49f-5(2,49)-3=-0,011751, 
¥"=3(249)2-5=13,6003, 

?-  =  3(2,49)  =  7,47. 


Newton's  method  of  approximation.  351 

But  the  equation  involving  u'  may  be  written  thus  : 
,         X."       Z"     ,^      1     ,3 

And  since  one  of  the  values  of  u'  must  be  less  than^^,  the 

1  1 

corresponding  values  of  u'^,  M'^  are  less  than   j^^qq-'    ioqqooo  ' 

X"  1 

hence  — ^^7,  will  represent  the  value  of  m'  to  within———. 

Since  we  have 

X"      0,011751         11751         ^^^^^ 

=  — =z =  0,0008  .  .  ., 

Y"         13,6003     13600300       ' 

it  follows  that  m'= 0,0008,  to  within  ,  and  consequently 

a;=2,49+0,0008=2,4908,  to  within 


10000 


Again,  by  supposing  a;= 2,4908 +m",  we  could  obtain  a  value  of 
1 


X  to  within 


100000000 


Each  operation  commonly  gives  the  root  to  twice  as  many  places 
of  decimals  as  the  previous  operation. 

310.  Generally,  let;?  andjp+1  be  two  numbers  between  which 
one  of  the  roots  of  the  equation  X=0  is  comprised. 

First  determine  the  value  of  this  root  to  loithin  — ,  by  substituting 

a  series  of  numbers   comprised  between  p  and  p+1,   until   two 
numbers  are  obtained  which  do  not  differ  from  each  other  by  more 

than  -. 


352 


Then,   calling  x'  the   value  of  x  obtained   to  within  —  suppose 

x=ix'-\-u  in  the  equation  X  =  0  ; 
which  gives 


Z' 

X'+Y'W  +  — M^  .   .   .    +M'»=:0: 


whicJi  can  he  put  under  the  form 


X',  Y'  Z'  .  .  .  being  easily  calculated.  (Art.  309). 

Since  the  sum  of  the  terms,  which  follow  —:^,  in  the  second  mem- 
ber of  this  equation  is,  commonly,  less  than  ,  they  can  be 
neglected,  and  calculating  —  ^7  to  within          ,  we  add  the  result  to 

x',  which  gives  a  new  value  x"  approximating  to  within  of  the 

exact  value. 

To  obtain  a  3d  approximation,  we  suppose  x=x"-\-u'  in  the  pro- 
posed equation,  which  gives 

Z" 

X'+Y'V  +  ym'^+  .  .  .m''"=0; 

X"       Z"     ,,  1 

whence  rt  =  — :r— , — ——-u'^—  .  .  .  _-—«''*. 

Z"  1 

Neglecting  the  terms  — --tttjm'-—  .  .  .  ——,%''"  which  are suppo- 

sed  to  be  less  than  0,0001,  we  calculate  the  value  of  — t?7,j  continuing 


Newton's  jiethod  of  approximation.  353 

the  operation  to  the  place  of  decimals,  and  add  the  result  to 

1 

x" ;  this  gives  a  third  approximation  x'",  exact  to  within  —         • 

Repeat  this  series  of  operations  for  each  of  the  positive  roots. 
As  for  the  negative  roots,  they  are  found  m  the  same  way  as  the 
positive  roots,  by  changing  x  into  —x  in  the  proposed  equation, 
which  then  becomes, 

_a;3_|_5_^,_3_0^  or  x^  —  5x+S=0 
in  which  the  positive  roots  taken  with  a  negative  sign,  are  the  nega- 
tive roots  of  the  proposed  equation.     These  roots  are     . 

a;=  — 1,8342  and  ic— =  0,6566 
to  within  0,0001. 


.\^ 


«v  > 


ii^^a>' 


\ 


« 


I 


